mmtheorems16 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19465

Type

Label

Description

Statement

Theorem

ecase23d 1501

Deduction for elimination by cases.

|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)

Theorem

3ecase 1502

Inference for elimination by cases.

|- (-. ph -> th)   &   |- (-. ps -> th)   &   |- (-. ch -> th)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th

Other axiomatizations of classical propositional calculus

Derive the Lukasiewicz axioms from Meredith's sole axiom

Theorem

meredith 1503

Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 1519, luk-2 1520, and luk-3 1521. Using these we finally re-derive our axioms as ax1 1530, ax2 1531, and ax3 1532, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (The proof was shortened by Andrew Salmon, 25-Jul-2011.) (The proof was shortened by Wolf Lammen, 28-May-2013.)

|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))

Theorem

meredithOLD 1504

Obsolete proof of meredith 1503. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))

Theorem

meredithOLDOLD 1505

Obsolete proof of meredith 1503.

|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))

Theorem

merlem1 1506

Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)

|- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))

Theorem

merlem2 1507

Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (((ph -> ph) -> ch) -> (th -> ch))

Theorem

merlem3 1508

Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (((ps -> ch) -> ph) -> (ch -> ph))

Theorem

merlem4 1509

Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (ta -> ((ta -> ph) -> (th -> ph)))

Theorem

merlem5 1510

Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- ((ph -> ps) -> (-. -. ph -> ps))

Theorem

merlem6 1511

Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (ch -> (((ps -> ch) -> ph) -> (th -> ph)))

Theorem

merlem7 1512

Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))

Theorem

merlem8 1513

Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))

Theorem

merlem9 1514

Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (((ph -> ps) -> (ch -> (th -> (ps -> ta)))) -> (et -> (ch -> (th -> (ps -> ta)))))

Theorem

merlem10 1515

Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))

Theorem

merlem11 1516

Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- ((ph -> (ph -> ps)) -> (ph -> ps))

Theorem

merlem12 1517

Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- (((th -> (-. -. ch -> ch)) -> ph) -> ph)

Theorem

merlem13 1518

Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

|- ((ph -> ps) -> (((th -> (-. -. ch -> ch)) -> -. -. ph) -> ps))

Theorem

luk-1 1519

1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.

|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))

Theorem

luk-2 1520

2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.

|- ((-. ph -> ph) -> ph)

Theorem

luk-3 1521

3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.

|- (ph -> (-. ph -> ps))

Derive the standard axioms from the Lukasiewicz axioms

Theorem

luklem1 1522

Used to rederive standard propositional axioms from Lukasiewicz'.

|- (ph -> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)

Theorem

luklem2 1523

Used to rederive standard propositional axioms from Lukasiewicz'.

|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))

Theorem

luklem3 1524

Used to rederive standard propositional axioms from Lukasiewicz'.

|- (ph -> (((-. ph -> ps) -> ch) -> (th -> ch)))

Theorem

luklem4 1525

Used to rederive standard propositional axioms from Lukasiewicz'.

|- ((((-. ph -> ph) -> ph) -> ps) -> ps)

Theorem

luklem5 1526

Used to rederive standard propositional axioms from Lukasiewicz'.

|- (ph -> (ps -> ph))

Theorem

luklem6 1527

Used to rederive standard propositional axioms from Lukasiewicz'.

|- ((ph -> (ph -> ps)) -> (ph -> ps))

Theorem

luklem7 1528

Used to rederive standard propositional axioms from Lukasiewicz'.

|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))

Theorem

luklem8 1529

Used to rederive standard propositional axioms from Lukasiewicz'.

|- ((ph -> ps) -> ((ch -> ph) -> (ch -> ps)))

Theorem

ax1 1530

Standard propositional axiom derived from Lukasiewicz axioms.

|- (ph -> (ps -> ph))

Theorem

ax2 1531

Standard propositional axiom derived from Lukasiewicz axioms.

|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))

Theorem

ax3 1532

Standard propositional axiom derived from Lukasiewicz axioms.

|- ((-. ph -> -. ps) -> (ps -> ph))

Logical 'nand' (Sheffer stroke)

Syntax

wnand 1533

Extend wff definition to include 'nand'.

wff (ph -/\ ps)

Definition

df-nand 1534

Define incompatibility ('not-and' or 'nand'). This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar.

|- ((ph -/\ ps) <-> -. (ph /\ ps))

Derive Nicod's axiom from the standard axioms

Theorem

nic-justlem 1535

Lemma for handling nested 'nand's. [Auxiliary lemma - not displayed.]

Theorem

nic-justim 1536

Show equivalence between implication and the Nicod version. To derive nic-dfim 1539, apply nic-justbi 1538.

|- ((ph -> ps) <-> (ph -/\ (ps -/\ ps)))

Theorem

nic-justneg 1537

Show equivalence between negation and the Nicod version. To derive nic-dfneg 1540, apply nic-justbi 1538.

|- (-. ps <-> (ps -/\ ps))

Theorem

nic-justbi 1538

Show equivalence between the bidirectional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.)

|- ((ph <-> ps) <-> ((ph -/\ ps) -/\ ((ph -/\ ph) -/\ (ps -/\ ps))))

Theorem

nic-dfim 1539

Define implication in terms of 'nand'. Analogous to ((ph -/\ (ps -/\ ps)) <-> (ph -> ps)). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement).

|- (((ph -/\ (ps -/\ ps)) -/\ (ph -> ps)) -/\ (((ph -/\ (ps -/\ ps)) -/\ (ph -/\ (ps -/\ ps))) -/\ ((ph -> ps) -/\ (ph -> ps))))

Theorem

nic-dfneg 1540

Define negation in terms of 'nand'. Analogous to ((ph -/\ ph) <-> -. ph). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement).

|- (((ph -/\ ph) -/\ -. ph) -/\ (((ph -/\ ph) -/\ (ph -/\ ph)) -/\ (-. ph -/\ -. ph)))

Theorem

nic-mp 1541

Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply ch, this form is necessary for useful derivations from nic-ax 1543. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.)

|- ph   &   |- (ph -/\ (ch -/\ ps))   =>   |- ps

Theorem

nic-mpALT 1542

A direct proof of nic-mp 1541.

|- ph   &   |- (ph -/\ (ch -/\ ps))   =>   |- ps

Theorem

nic-ax 1543

Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1503, the usual axioms can be derived from this and vice versa. Unlike meredith 1503, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1543, nic-mp 1541 } is equivalent to { luk-1 1519, luk-2 1520, luk-3 1521, ax-mp 7 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.)

|- ((ph -/\ (ch -/\ ps)) -/\ ((ta -/\ (ta -/\ ta)) -/\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))

Theorem

nic-axALT 1544

A direct proof of nic-ax 1543.

|- ((ph -/\ (ch -/\ ps)) -/\ ((ta -/\ (ta -/\ ta)) -/\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))

Derive the Lukasiewicz axioms from Nicod's axiom

Theorem

nic-imp 1545

Inference for nic-mp 1541 using nic-ax 1543 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ph -/\ (ch -/\ ps))   =>   |- ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))

Theorem

nic-idlem1 1546

Lemma for nic-id 1548. [Auxiliary lemma - not displayed.]

Theorem

nic-idlem2 1547

Lemma for nic-id 1548. Inference used by nic-id 1548. [Auxiliary lemma - not displayed.]

Theorem

nic-id 1548

Theorem id 15 expressed with -/\. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ta -/\ (ta -/\ ta))

Theorem

nic-swap 1549

-/\ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- ((th -/\ ph) -/\ ((ph -/\ th) -/\ (ph -/\ th)))

Theorem

nic-isw1 1550

Inference version of nic-swap 1549. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (th -/\ ph)   =>   |- (ph -/\ th)

Theorem

nic-isw2 1551

Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ps -/\ (th -/\ ph))   =>   |- (ps -/\ (ph -/\ th))

Theorem

nic-iimp1 1552

Inference version of nic-imp 1545 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ph -/\ (ch -/\ ps))   &   |- (th -/\ ch)   =>   |- (th -/\ ph)

Theorem

nic-iimp2 1553

Inference version of nic-imp 1545 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- ((ph -/\ ps) -/\ (ch -/\ ch))   &   |- (th -/\ ph)   =>   |- (th -/\ (ch -/\ ch))

Theorem

nic-idel 1554

Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ph -/\ (ch -/\ ps))   =>   |- (ph -/\ (ch -/\ ch))

Theorem

nic-ich 1555

Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ph -/\ (ps -/\ ps))   &   |- (ps -/\ (ch -/\ ch))   =>   |- (ph -/\ (ch -/\ ch))

Theorem

nic-idbl 1556

Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.)

|- (ph -/\ (ps -/\ ps))   =>   |- ((ps -/\ ps) -/\ ((ph -/\ ph) -/\ (ph -/\ ph)))

Theorem

nic-bijust 1557

For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1558 and nic-bi2 1559 are used to convert the definitions into usable theorems about one side of the implication. (Contributed by Jeff Hoffman, 18-Nov-2007.)

|- ((ta -/\ ta) -/\ ((ta -/\ ta) -/\ (ta -/\ ta)))

Theorem

nic-bi1 1558

Inference to extract one side of an implication from a definition.

|- ((ph -/\ ps) -/\ ((ph -/\ ph) -/\ (ps -/\ ps)))   =>   |- (ph -/\ (ps -/\ ps))

Theorem

nic-bi2 1559

Inference to extract the other side of an implication from a 'biconditional' definition.

|- ((ph -/\ ps) -/\ ((ph -/\ ph) -/\ (ps -/\ ps)))   =>   |- (ps -/\ (ph -/\ ph))

Theorem

nic-stdmp 1560

Derive the standard modus ponens from nic-mp 1541. (Contributed by Jeff Hoffman, 18-Nov-2007.)

|- ph   &   |- (ph -> ps)   =>   |- ps

Theorem

nic-luk1 1561

Proof of luk-1 1519 from nic-ax 1543 and nic-mp 1541 (and definitions nic-dfim 1539 and nic-dfneg 1540). Note that the standard axioms ax-1 4, ax-2 5, and ax-3 6 are proved from the Lukasiewicz axioms by theorems ax1 1530, ax2 1531, and ax3 1532. (Contributed by Jeff Hoffman, 18-Nov-2007.)

|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))

Theorem

nic-luk2 1562

Proof of luk-2 1520 from nic-ax 1543 and nic-mp 1541. (Contributed by Jeff Hoffman, 18-Nov-2007.)

|- ((-. ph -> ph) -> ph)

Theorem

nic-luk3 1563

Proof of luk-3 1521 from nic-ax 1543 and nic-mp 1541. (Contributed by Jeff Hoffman, 18-Nov-2007.)

|- (ph -> (-. ph -> ps))

True and false constants

Syntax

wtru 1564

T. is a wff.

wff T.

Syntax

wfal 1565

F. is a wff.

wff F.

Definition

df-tru 1566

Definition of T., a tautology. T. is a constant true. In this definition biid 289 is used as an antecedent, however, any true wff, such as an axiom, can be used in its place.

|- ( T. <-> (ph <-> ph))

Definition

df-fal 1567

Definition of F., a contradiction. F. is a constant false.

|- ( F. <-> -. T. )

Theorem

tru 1568

T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

|- T.

Theorem

notfal 1569

F. is not provable. (Contributed by Anthony Hart, 22-Oct-2010.) (The proof was shortened by Mel L. O'Cat, 11-Mar-2012.)

|- -. F.

Auxiliary theorems for Alan Sare's virtual deduction, part 1

Theorem

iin3 1570

in3 17600 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.)

|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> (ps -> (ch -> th)))

Theorem

iidn3 1571

idn3 17603 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.)

|- (ph -> (ps -> (ch -> ch)))

Theorem

ee222 1572

e222 17623 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.)

|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   &   |- (ph -> (ps -> ta))   &   |- (ch -> (th -> (ta -> et)))   =>   |- (ph -> (ps -> et))

Theorem

ee22 1573

Virtual deduction rule e22 17658 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 2-May-2011.)

|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   &   |- (ch -> (th -> ta))   =>   |- (ph -> (ps -> ta))

Theorem

ee12an 1574

e12an 17689 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.)

|- (ph -> ps)   &   |- (ph -> (ch -> th))   &   |- ((ps /\ th) -> ta)   =>   |- (ph -> (ch -> ta))

Theorem

ee3bir 1575

Right-biconditional form of e3 17700 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.)

|- (ph -> (ps -> (ch -> th)))   &   |- (ta <-> th)   =>   |- (ph -> (ps -> (ch -> ta)))

Theorem

ee13 1576

e13 17711 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.)

|- (ph -> ps)   &   |- (ph -> (ch -> (th -> ta)))   &   |- (ps -> (ta -> et))   =>   |- (ph -> (ch -> (th -> et)))

Theorem

ee23 1577

e23 17718 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.)

|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> (th -> ta)))   &   |- (ch -> (ta -> et))   =>   |- (ph -> (ps -> (th -> et)))

Theorem

ee121 1578

e121 17643 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.)

|- (ph -> ps)   &   |- (ph -> (ch -> th))   &   |- (ph -> ta)   &   |- (ps -> (th -> (ta -> et)))   =>   |- (ph -> (ch -> et))

Theorem

ee122 1579

e122 17640 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.)

|- (ph -> ps)   &   |- (ph -> (ch -> th))   &   |- (ph -> (ch -> ta))   &   |- (ps -> (th -> (ta -> et)))   =>   |- (ph -> (ch -> et))

Theorem

ee333 1580

e333 17696 without virtual deductions.

|- (ph -> (ps -> (ch -> th)))   &   |- (ph -> (ps -> (ch -> ta)))   &   |- (ph -> (ps -> (ch -> et)))   &   |- (th -> (ta -> (et -> ze)))   =>   |- (ph -> (ps -> (ch -> ze)))

Theorem

ee323 1581

e323 17728 without virtual deductions.

|- (ph -> (ps -> (ch -> th)))   &   |- (ph -> (ps -> ta))   &   |- (ph -> (ps -> (ch -> et)))   &   |- (th -> (ta -> (et -> ze)))   =>   |- (ph -> (ps -> (ch -> ze)))

Theorem

3ornot23 1582

If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 17766. (Contributed by Alan Sare, 31-Dec-2011.)

|- ((-. ph /\ -. ps) -> ((ch \/ ph \/ ps) -> ch))

Theorem

orbi1r 1583

orbi1 817 with order of disjuncts reversed. Derived from orbi1rVD 17767. (Contributed by Alan Sare, 31-Dec-2011.)

|- ((ph <-> ps) -> ((ch \/ ph) <-> (ch \/ ps)))

Theorem

bitr3 1584

Closed nested implication form of bitr3i 309. Derived automatically from bitr3VD 17768. (Contributed by Alan Sare, 31-Dec-2011.)

|- ((ph <-> ps) -> ((ph <-> ch) -> (ps <-> ch)))

Theorem

3orbi123 1585

pm4.39 994 with a 3-conjunct antecedent. This proof is 3orbi123VD 17769 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011).

|- (((ph <-> ps) /\ (ch <-> th) /\ (ta <-> et)) -> ((ph \/ ch \/ ta) <-> (ps \/ th \/ et)))

Theorem

exbir 1586

Exportation implication also converting head from biconditional to conditional. This proof is exbirVD 17772 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.)

|- (((ph /\ ps) -> (ch <-> th)) -> (ph -> (ps -> (th -> ch))))

Theorem

3impexp 1587

impexp 500 with a 3-conjunct antecedent. This proof is 3impexpVD 17775 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.)

|- (((ph /\ ps /\ ch) -> th) <-> (ph -> (ps -> (ch -> th))))

Theorem

3impexpbicom 1588

3impexp 1587 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 17775. (Contributed by Alan Sare, 31-Dec-2011.)

|- (((ph /\ ps /\ ch) -> (th <-> ta)) <-> (ph -> (ps -> (ch -> (ta <-> th)))))

Theorem

3impexpbicomi 1589

Deduction form of 3impexpbicom 1588. Derived automatically from 3impexpbicomiVD 17777. (Contributed by Alan Sare, 31-Dec-2011.)

|- ((ph /\ ps /\ ch) -> (th <-> ta))   =>   |- (ph -> (ps -> (ch -> (ta <-> th))))

Theorem

syl5imp 1590

Closed form of syl5OLD 63. Derived automatically from syl5impVD 17782. (Contributed by Alan Sare, 31-Dec-2011.)

|- ((ph -> (ps -> ch)) -> ((th -> ps) -> (ph -> (th -> ch))))

Theorem

idi 1591

Deduction form of id 15. Derived automatically from idiVD 17783. (Contributed by Alan Sare, 31-Dec-2011.)

|- ph   =>   |- ph

Theorem

ancomsimp 1592

Closed form of ancoms 512. Derived automatically from ancomsimpVD 17784. (Contributed by Alan Sare, 31-Dec-2011.)

|- (((ph /\ ps) -> ch) <-> ((ps /\ ph) -> ch))

Theorem

impexp3a 1593

The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed it was minimized. The completed User's Proof before minimization is not shown.

1::	|- (((ps /\ ch) -> th) <-> (ps -> (ch -> th)))
qed:1:	|- ((ph -> ((ps /\ ch) -> th)) <-> (ph -> (ps -> (ch -> th))))

|- ((ph -> ((ps /\ ch) -> th)) <-> (ph -> (ps -> (ch -> th))))

Theorem

com3rgbi 1594

The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

1::	|- ((ph -> (ps -> (ch -> th))) -> (ph -> (ch -> (ps -> th))))
2::	|- ((ph -> (ch -> (ps -> th))) -> (ch -> (ph -> (ps -> th))))
3:1,2:	|- ((ph -> (ps -> (ch -> th))) -> (ch -> (ph -> (ps -> th))))
4::	|- ((ch -> (ph -> (ps -> th))) -> (ph -> (ch -> (ps -> th))))
5::	|- ((ph -> (ch -> (ps -> th))) -> (ph -> (ps -> (ch -> th))))
6:4,5:	|- ((ch -> (ph -> (ps -> th))) -> (ph -> (ps -> (ch -> th))))
qed:3,6:	|- ((ph -> (ps -> (ch -> th))) <-> (ch -> (ph -> (ps -> th))))

|- ((ph -> (ps -> (ch -> th))) <-> (ch -> (ph -> (ps -> th))))

Theorem

impexp3acom3r 1595

The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

1::	|- ((ph -> ((ps /\ ch) -> th)) <-> (ph -> (ps -> (ch -> th))))
2::	|- ((ps -> (ch -> (ph -> th))) <-> (ph -> (ps -> (ch -> th))))
qed:1,2:	|- ((ph -> ((ps /\ ch) -> th)) <-> (ps -> (ch -> (ph -> th))))

|- ((ph -> ((ps /\ ch) -> th)) <-> (ps -> (ch -> (ph -> th))))

Theorem

exp3acom3r 1596

The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	|- (ph -> ((ps /\ ch) -> th))
qed:1:	|- (ps -> (ch -> (ph -> th)))

|- (ph -> ((ps /\ ch) -> th))   =>   |- (ps -> (ch -> (ph -> th)))

Theorem

exp3acom23g 1597

Implication form of exp3acom23 1598.(Contributed by Alan Sare, 22-Jul-2012.)

|- ((ph -> ((ps /\ ch) -> th)) <-> (ph -> (ch -> (ps -> th))))

Theorem

exp3acom23 1598

The exportation deduction exp3a 496 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)

|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> (ps -> th)))

Theorem

simplbi2comg 1599

Implication form of simplbi2com 1600. (Contributed by Alan Sare, 22-Jul-2012.)

|- ((ph <-> (ps /\ ch)) -> (ch -> (ps -> ph)))

Theorem

simplbi2com 1600

A deduction eliminating a conjunct, similar to simplbi2 719. (Contributed by Alan Sare, 22-Jul-2012.) (The proof was shortened by Wolf Lammen, 10-Nov-2012.

|- (ph <-> (ps /\ ch))   =>   |- (ch -> (ps -> ph))