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Statement List for Metamath Proof Explorer - 1501-1600 - Page 16 of 195
TypeLabelDescription
Statement
 
Theoremecase23d 1501 Deduction for elimination by cases.
|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)
 
Theorem3ecase 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
 
Theoremmeredith 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)))
 
TheoremmeredithOLD 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)))
 
TheoremmeredithOLDOLD 1505 Obsolete proof of meredith 1503.
|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))
 
Theoremmerlem1 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))
 
Theoremmerlem2 1507 Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ph) -> ch) -> (th -> ch))
 
Theoremmerlem3 1508 Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> ph) -> (ch -> ph))
 
Theoremmerlem4 1509 Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ta -> ((ta -> ph) -> (th -> ph)))
 
Theoremmerlem5 1510 Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (-. -. ph -> ps))
 
Theoremmerlem6 1511 Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ch -> (((ps -> ch) -> ph) -> (th -> ph)))
 
Theoremmerlem7 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)))
 
Theoremmerlem8 1513 Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))
 
Theoremmerlem9 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)))))
 
Theoremmerlem10 1515 Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))
 
Theoremmerlem11 1516 Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
 
Theoremmerlem12 1517 Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((th -> (-. -. ch -> ch)) -> ph) -> ph)
 
Theoremmerlem13 1518 Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (((th -> (-. -. ch -> ch)) -> -. -. ph) -> ps))
 
Theoremluk-1 1519 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))
 
Theoremluk-2 1520 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((-. ph -> ph) -> ph)
 
Theoremluk-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
 
Theoremluklem1 1522 Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)
 
Theoremluklem2 1523 Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))
 
Theoremluklem3 1524 Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (((-. ph -> ps) -> ch) -> (th -> ch)))
 
Theoremluklem4 1525 Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((((-. ph -> ph) -> ph) -> ps) -> ps)
 
Theoremluklem5 1526 Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (ps -> ph))
 
Theoremluklem6 1527 Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
 
Theoremluklem7 1528 Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
 
Theoremluklem8 1529 Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> ps) -> ((ch -> ph) -> (ch -> ps)))
 
Theoremax1 1530 Standard propositional axiom derived from Lukasiewicz axioms.
|- (ph -> (ps -> ph))
 
Theoremax2 1531 Standard propositional axiom derived from Lukasiewicz axioms.
|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))
 
Theoremax3 1532 Standard propositional axiom derived from Lukasiewicz axioms.
|- ((-. ph -> -. ps) -> (ps -> ph))
 
Logical 'nand' (Sheffer stroke)
 
Syntaxwnand 1533 Extend wff definition to include 'nand'.
wff (ph -/\ ps)
 
Definitiondf-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
 
Theoremnic-justlem 1535 Lemma for handling nested 'nand's. [Auxiliary lemma - not displayed.]
 
Theoremnic-justim 1536 Show equivalence between implication and the Nicod version. To derive nic-dfim 1539, apply nic-justbi 1538.
|- ((ph -> ps) <-> (ph -/\ (ps -/\ ps)))
 
Theoremnic-justneg 1537 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1540, apply nic-justbi 1538.
|- (-. ps <-> (ps -/\ ps))
 
Theoremnic-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))))
 
Theoremnic-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))))
 
Theoremnic-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)))
 
Theoremnic-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
 
Theoremnic-mpALT 1542 A direct proof of nic-mp 1541.
|- ph   &   |- (ph -/\ (ch -/\ ps))   =>   |- ps
 
Theoremnic-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)))))
 
Theoremnic-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
 
Theoremnic-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)))
 
Theoremnic-idlem1 1546 Lemma for nic-id 1548. [Auxiliary lemma - not displayed.]
 
Theoremnic-idlem2 1547 Lemma for nic-id 1548. Inference used by nic-id 1548. [Auxiliary lemma - not displayed.]
 
Theoremnic-id 1548 Theorem id 15 expressed with -/\. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- (ta -/\ (ta -/\ ta))
 
Theoremnic-swap 1549 -/\ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- ((th -/\ ph) -/\ ((ph -/\ th) -/\ (ph -/\ th)))
 
Theoremnic-isw1 1550 Inference version of nic-swap 1549. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- (th -/\ ph)   =>   |- (ph -/\ th)
 
Theoremnic-isw2 1551 Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- (ps -/\ (th -/\ ph))   =>   |- (ps -/\ (ph -/\ th))
 
Theoremnic-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)
 
Theoremnic-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))
 
Theoremnic-idel 1554 Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- (ph -/\ (ch -/\ ps))   =>   |- (ph -/\ (ch -/\ ch))
 
Theoremnic-ich 1555 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|- (ph -/\ (ps -/\ ps))   &   |- (ps -/\ (ch -/\ ch))   =>   |- (ph -/\ (ch -/\ ch))
 
Theoremnic-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)))
 
Theoremnic-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)))
 
Theoremnic-bi1 1558 Inference to extract one side of an implication from a definition.
|- ((ph -/\ ps) -/\ ((ph -/\ ph) -/\ (ps -/\ ps)))   =>   |- (ph -/\ (ps -/\ ps))
 
Theoremnic-bi2 1559 Inference to extract the other side of an implication from a 'biconditional' definition.
|- ((ph -/\ ps) -/\ ((ph -/\ ph) -/\ (ps -/\ ps)))   =>   |- (ps -/\ (ph -/\ ph))
 
Theoremnic-stdmp 1560 Derive the standard modus ponens from nic-mp 1541. (Contributed by Jeff Hoffman, 18-Nov-2007.)
|- ph   &   |- (ph -> ps)   =>   |- ps
 
Theoremnic-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)))
 
Theoremnic-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)
 
Theoremnic-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
 
Syntaxwtru 1564 T. is a wff.
wff T.
 
Syntaxwfal 1565 F. is a wff.
wff F.
 
Definitiondf-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))
 
Definitiondf-fal 1567 Definition of F., a contradiction. F. is a constant false.
|- ( F. <-> -. T. )
 
Theoremtru 1568 T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
 
Theoremnotfal 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
 
Theoremiin3 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)))
 
Theoremiidn3 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)))
 
Theoremee222 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))
 
Theoremee22 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))
 
Theoremee12an 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))
 
Theoremee3bir 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)))
 
Theoremee13 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)))
 
Theoremee23 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)))
 
Theoremee121 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))
 
Theoremee122 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))
 
Theoremee333 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)))
 
Theoremee323 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)))
 
Theorem3ornot23 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))
 
Theoremorbi1r 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)))
 
Theorembitr3 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)))
 
Theorem3orbi123 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)))
 
Theoremexbir 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))))
 
Theorem3impexp 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))))
 
Theorem3impexpbicom 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)))))
 
Theorem3impexpbicomi 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))))
 
Theoremsyl5imp 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))))
 
Theoremidi 1591 Deduction form of id 15. Derived automatically from idiVD 17783. (Contributed by Alan Sare, 31-Dec-2011.)
|- ph   =>   |- ph
 
Theoremancomsimp 1592 Closed form of ancoms 512. Derived automatically from ancomsimpVD 17784. (Contributed by Alan Sare, 31-Dec-2011.)
|- (((ph /\ ps) -> ch) <-> ((ps /\ ph) -> ch))
 
Theoremimpexp3a 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))))
 
Theoremcom3rgbi 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))))
 
Theoremimpexp3acom3r 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))))
 
Theoremexp3acom3r 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)))
 
Theoremexp3acom23g 1597 Implication form of exp3acom23 1598.(Contributed by Alan Sare, 22-Jul-2012.)
|- ((ph -> ((ps /\ ch) -> th)) <-> (ph -> (ch -> (ps -> th))))
 
Theoremexp3acom23 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)))
 
Theoremsimplbi2comg 1599 Implication form of simplbi2com 1600. (Contributed by Alan Sare, 22-Jul-2012.)
|- ((ph <-> (ps /\ ch)) -> (ch -> (ps -> ph)))
 
Theoremsimplbi2com 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))

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