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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremequviniNEW7 29601 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

TheoremequveliNEW7 29602 A variable elimination law for equality with no distinct variable requirements. (Compare equviniNEW7 29601.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29516 instead of ax-7 1750.

TheoremequvinNEW7 29603* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Theoremax11v2NEW7 29604* Recovery of ax-11o 2220 from ax11v 2174. This proof uses ax-10 2219 and ax-11 1762. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11a2NEW7 29605* Derive ax-11o 2220 from a hypothesis in the form of ax-11 1762. ax-10 2219 and ax-11 1762 are used by the proof, but not ax-10o 2218 or ax-11o 2220. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11oNEW7 29606 Derivation of set.mm's original ax-11o 2220 from ax-10 2219 and the shorter ax-11 1762 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2223 or ax-17 1627 (given all of the original and new versions of sp 1764 through ax-15 2222).

Another open problem is whether this theorem can be proved without relying on ax12o 2011.

Theorem ax11 2234 shows the reverse derivation of ax-11 1762 from ax-11o 2220.

Normally, ax11o 2082 should be used rather than ax-11o 2220, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Theoremequs4NEW7 29607 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequs5NEW7 29608 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremequs5bAUX7 29609 Lemma used in proofs of substitution properties. (Contributed by NM, 27-Oct-2017.)

Theoremax15NEW7 29610 Axiom ax-15 2222 is redundant if we assume ax-17 1627. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2108 and ax-17 1627. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.)

Theoremsb2NEW7 29611 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremequsb1NEW7 29612 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2NEW7 29613 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

TheoremsbiedNEW7 29614 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2152). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

TheoremsbieNEW7 29615 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremhbsb2aNEW7 29616 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2eNEW7 29617 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3NEW7 29618 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1NEW7 29619 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorema16gNEW7 29620* Generalization of ax16 2051. (Contributed by NM, 25-Jul-2015.)

Theorema16gbNEW7 29621* A generalization of axiom ax-16 2223. (Contributed by NM, 5-Aug-1993.)

Theorema16nfwAUX7 29622* Weak version of a16nf 2054 not requiring ax-7 1750. (Contributed by NM, 10-Oct-2017.)

Theoremax16NEW7 29623* Proof of older axiom ax-16 2223. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)

Theorema16nfNEW7 29624* If dtru 4393 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29516 instead of ax-7 1750.

Theoremax16iNEW7 29625* Inference with ax16 2051 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

Theoremsb4NEW7 29626 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4bNEW7 29627 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Theoremhbsb2NEW7 29628 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4NEW7 29629 The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. See also spsbc 3175 and rspsbc 3241. (Contributed by NM, 5-Aug-1993.)

TheoremsbftNEW7 29630 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)

TheoremsbfNEW7 29631 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbequ5wAUX7 29632* Weak version of sbequ5 2126 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

TheoremsbhNEW7 29633 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Theoremsbf2NEW7 29634 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremnfsb2NEW7 29635 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)

TheoremsbnNEW7 29636 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsbi1NEW7 29637 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbi2NEW7 29638 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)

TheoremsbimNEW7 29639 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

TheoremsborNEW7 29640 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)

TheoremsbrimNEW7 29641 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

TheoremsblimNEW7 29642 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)

TheoremsbanNEW7 29643 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

TheoremsbbiNEW7 29644 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

TheoremspsbeNEW7 29645 A specialization theorem. (Contributed by NM, 5-Aug-1993.)

TheoremspsbimNEW7 29646 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

TheoremspsbbiNEW7 29647 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)

TheoremsbbidNEW7 29648 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ8NEW7 29649 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)

Theoremnfsb4twAUX7 29650* Weak version of nfsb4t 2129 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

Theoremnfsb4wAUX7 29651* Weak version of nfsb4t 2129 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

TheoremnfsbdwAUX7 29652* Deduction version of nfsbwAUX7 29739. (Contributed by NM, 24-May-2018.)

TheoremsbequiNEW7 29653 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

TheoremsbequNEW7 29654 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremdrsb2NEW7 29655 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

TheoremsbcoNEW7 29656 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2NEW7 29657 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

TheoremsbidmNEW7 29658 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbco2wAUX7 29659* Weak version of sbco2 2163 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

Theoremsbco2dwAUX7 29660* Weak version of sbco2d 2164 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

Theoremsbco3wAUX7 29661* Weak version of sbco3 2165 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

TheoremsbcomwAUX7 29662* Weak version of sbcom 2166 not requiring ax-7 1750. (Contributed by NM, 27-Oct-2017.)

Theoremsb8iwAUX7 29663* Version of sb8 2170 not requiring ax-7 1750. (Contributed by NM, 22-May-2018.)

Theoremsb8dwAUX7 29664* Weak version of sb8 2170 not requiring ax-7 1750. (Contributed by NM, 22-May-2018.)

Theoremsb5rfNEW7 29665 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb6rfNEW7 29666 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb8wAUX7 29667* Weak version of sb8 2170 not requiring ax-7 1750. (Contributed by NM, 28-Oct-2017.)

Theoremsb8ewAUX7 29668* Weak version of sb8e 2171 not requiring ax-7 1750. (Contributed by NM, 28-Oct-2017.)

Theoremax11vNEW7 29669* This is a version of ax-11o 2220 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2079 for the rederivation of ax-11o 2220 from this theorem. (Contributed by NM, 5-Aug-1993.)

Theoremsb56NEW7 29670* Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1660. (Contributed by NM, 14-Apr-2008.)

Theoremsb6NEW7 29671* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.)

Theoremsb5NEW7 29672* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)

TheoremexsbNEW7 29673* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) (Revised by NM, 28-Nov-2017.) Revised to prove from ax-7v 29516 instead of ax-7 1750.

TheoremexsbOLDNEW7 29674* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.)

Theoremequsb3lemNEW7 29675* Lemma for equsb3 2180. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremequsb3NEW7 29676* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Theoremelsb3NEW7 29677* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremelsb4NEW7 29678* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremhbs1NEW7 29679* is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremnfs1vNEW7 29680* is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)

TheoremsbhbwAUX7 29681* Weak version of sbhb 2185 not requiring ax-7 1750. (Contributed by NM, 28-Oct-2017.)

Theoremsbnf2NEW7 29682* Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theorem2sb5NEW7 29683* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theorem2sb6NEW7 29684* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theoremsb6aNEW7 29685* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2vNEW7 29686* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

TheoremsbelxNEW7 29687* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbel2xNEW7 29688* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbal1NEW7 29689* A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.)

TheoremsbalNEW7 29690* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)

TheoremsbexNEW7 29691* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)

TheoremsbalvNEW7 29692* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

TheoremnaecomsNEW7 29693 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

TheoremchvarNEW7 29694 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequs45fNEW7 29695 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremax11bNEW7 29696 A bidirectional version of ax11oNEW7 29606. (Contributed by NM, 30-Jun-2006.)

TheoremspvNEW7 29697* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)

TheoremspimevNEW7 29698* Distinct-variable version of spimeNEW7 29583. (Contributed by NM, 5-Aug-1993.)

TheoremspeivNEW7 29699* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)

TheoremchvarvNEW7 29700* Implicit substitution of for into a theorem. (Contributed by NM, 20-Apr-1994.)

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