HomeHome Intuitionistic Logic Explorer
Theorem List (p. 18 of 106)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb5f 1701 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremsb4e 1702 One direction of a simplified definition of substitution that unlike sb4 1729 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph )
 )
 
Theoremhbsb2a 1703 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
 
Theoremhbsb2e 1704 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )
 
Theoremhbsb3 1705 If  y is not free in  ph,  x is not free in  [
y  /  x ] ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1 1706 If  y is not free in  ph,  x is not free in  [
y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ y ph   =>    |- 
 F/ x [ y  /  x ] ph
 
Theoremsbcof2 1707 Version of sbco 1858 where  x is not free in  ph. (Contributed by Jim Kingdon, 28-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
1.4  Predicate calculus with distinct variables
 
1.4.1  Derive the axiom of distinct variables ax-16
 
Theoremspimv 1708* A version of spim 1642 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  ps )
 
Theoremaev 1709* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1711. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremax16 1710* Theorem showing that ax-16 1711 is redundant if ax-17 1435 is included in the axiom system. The important part of the proof is provided by aev 1709.

See ax16ALT 1755 for an alternate proof that does not require ax-10 1412 or ax-12 1418.

This theorem should not be referenced in any proof. Instead, use ax-16 1711 below so that theorems needing ax-16 1711 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Axiomax-16 1711* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1435 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory, but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1435; see theorem ax16 1710.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1710. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremdveeq2 1712* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremdveeq2or 1713* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1712 but connecting  A. x x  =  y by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
 |-  ( A. x  x  =  y  \/  F/ x  z  =  y
 )
 
TheoremdvelimfALT2 1714* Proof of dvelimf 1907 using dveeq2 1712 (shown as the last hypothesis) instead of ax-12 1418. This shows that ax-12 1418 could be replaced by dveeq2 1712 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremnd5 1715* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
 |-  ( -.  A. y  y  =  x  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremexlimdv 1716* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch ) )
 
Theoremax11v2 1717* Recovery of ax11o 1719 from ax11v 1724 without using ax-11 1413. The hypothesis is even weaker than ax11v 1724, with  z both distinct from  x and not occurring in  ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1719. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11a2 1718* Derive ax-11o 1720 from a hypothesis in the form of ax-11 1413. The hypothesis is even weaker than ax-11 1413, with  z both distinct from  x and not occurring in  ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1719. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( A. z ph  ->  A. x ( x  =  z  ->  ph )
 ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
1.4.2  Derive the obsolete axiom of variable substitution ax-11o
 
Theoremax11o 1719 Derivation of set.mm's original ax-11o 1720 from the shorter ax-11 1413 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1711 or ax-17 1435.

Normally, ax11o 1719 should be used rather than ax-11o 1720, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Axiomax-11o 1720 Axiom ax-11o 1720 ("o" for "old") was the original version of ax-11 1413, before it was discovered (in Jan. 2007) that the shorter ax-11 1413 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of " -.  A. x x  =  y  ->..." as informally meaning "if  x and  y are distinct variables then..." The antecedent becomes false if the same variable is substituted for  x and  y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form  -.  A. x x  =  y a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1719.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1719. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
1.4.3  More theorems related to ax-11 and substitution
 
Theoremalbidv 1721* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidv 1722* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremax11b 1723 A bidirectional version of ax-11o 1720. (Contributed by NM, 30-Jun-2006.)
 |-  ( ( -.  A. x  x  =  y  /\  x  =  y
 )  ->  ( ph  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11v 1724* This is a version of ax-11o 1720 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax11ev 1725* Analogue to ax11v 1724 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
 |-  ( x  =  y 
 ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
 
Theoremequs5 1726 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremequs5or 1727 Lemma used in proofs of substitution properties. Like equs5 1726 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb3 1728 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
 
Theoremsb4 1729 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4or 1730 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1729 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4b 1731 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb4bor 1732 Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
 |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremhbsb2 1733 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph ) )
 
Theoremnfsb2or 1734 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1733 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  F/ x [ y  /  x ] ph )
 
Theoremsbequilem 1735 Propositional logic lemma used in the sbequi 1736 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
 |-  ( ph  \/  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( ta  \/  ( ps  ->  ( th  ->  et )
 ) )   =>    |-  ( ph  \/  ( ta  \/  ( ps  ->  ( ch  ->  et )
 ) ) )
 
Theoremsbequi 1736 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
Theoremsbequ 1737 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.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremdrsb2 1738 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.)
 |-  ( A. x  x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremspsbe 1739 A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 1740 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 )
 
Theoremspsbbi 1741 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsbbid 1742 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch ) )
 
Theoremsbequ8 1743 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremsbft 1744 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
Theoremsbid2h 1745 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbid2 1746 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbidm 1747 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
 |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsb5rf 1748 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( ph  <->  E. y ( y  =  x  /\  [
 y  /  x ] ph ) )
 
Theoremsb6rf 1749 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
 )
 
Theoremsb8h 1750 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8eh 1751 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremsb8 1752 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
 |- 
 F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8e 1753 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
 |- 
 F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
1.4.4  Predicate calculus with distinct variables (cont.)
 
Theoremax16i 1754* Inference with ax-16 1711 as its conclusion, that doesn't require ax-10 1412, ax-11 1413, or ax-12 1418 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theoremax16ALT 1755* Version of ax16 1710 that doesn't require ax-10 1412 or ax-12 1418 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremspv 1756* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspimev 1757* Distinct-variable version of spime 1645. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspeiv 1758* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
Theoremequvin 1759* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theorema16g 1760* A generalization of axiom ax-16 1711. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theorema16gb 1761* A generalization of axiom ax-16 1711. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 1762* If there is only one element in the universe, then everything satisfies  F/. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theorem2albidv 1763* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps 
 <-> 
 A. x A. y ch ) )
 
Theorem2exbidv 1764* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps 
 <-> 
 E. x E. y ch ) )
 
Theorem3exbidv 1765* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
 
Theorem4exbidv 1766* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z E. w ps  <->  E. x E. y E. z E. w ch ) )
 
Theorem19.9v 1767* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
 |-  ( E. x ph  <->  ph )
 
Theoremexlimdd 1768 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  E. x ps )   &    |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theorem19.21v 1769* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  ( ph  ->  A. x ph ) in 19.21 1491 via the use of distinct variable conditions combined with ax-17 1435. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 1921 derived from df-eu 1919. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremalrimiv 1770* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalrimivv 1771* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x A. y ps )
 
Theoremalrimdv 1772* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch ) )
 
Theoremnfdv 1773* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )   =>    |-  ( ph  ->  F/ x ps )
 
Theorem2ax17 1774* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
 |-  ( ph  ->  A. x A. y ph )
 
Theoremalimdv 1775* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  A. x ch ) )
 
Theoremeximdv 1776* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theorem2alimdv 1777* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps  ->  A. x A. y ch ) )
 
Theorem2eximdv 1778* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  E. x E. y ch ) )
 
Theorem19.23v 1779* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
 |-  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theorem19.23vv 1780* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x E. y ph  ->  ps )
 )
 
Theoremsb56 1781* Two equivalent ways of expressing the proper substitution of  y for  x in  ph, when  x and  y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1662. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb6 1782* 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.) (Revised by NM, 14-Apr-2008.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5 1783* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremsbnv 1784* Version of sbn 1842 where  x and  y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbanv 1785* Version of sban 1845 where  x and  y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
 |-  ( [ y  /  x ] ( ph  /\  ps ) 
 <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps ) )
 
Theoremsborv 1786* Version of sbor 1844 where  x and  y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
 |-  ( [ y  /  x ] ( ph  \/  ps )  <->  ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) )
 
Theoremsbi1v 1787* Forward direction of sbimv 1789. (Contributed by Jim Kingdon, 25-Dec-2017.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2v 1788* Reverse direction of sbimv 1789. (Contributed by Jim Kingdon, 18-Jan-2018.)
 |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
Theoremsbimv 1789* Intuitionistic proof of sbim 1843 where  x and  y are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsblimv 1790* Version of sblim 1847 where  x and  y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps ) )
 
Theorempm11.53 1791* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph 
 ->  A. y ps )
 )
 
Theoremexlimivv 1792* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  ps )
 
Theoremexlimdvv 1793* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  ch ) )
 
Theoremexlimddv 1794* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theorem19.27v 1795* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28v 1796* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.36aiv 1797* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem19.41v 1798* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.41vv 1799* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( E. x E. y ph  /\  ps )
 )
 
Theorem19.41vvv 1800* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  ( E. x E. y E. z ph  /\ 
 ps ) )
    < Previous  Next >

Page List
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 10501-10511
  Copyright terms: Public domain < Previous  Next >