HomeHome Intuitionistic Logic Explorer
Theorem List (p. 17 of 133)
< 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 - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.29r2 1601 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
 |-  ( ( E. x E. y ph  /\  A. x A. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.29x 1602 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
 |-  ( ( E. x A. y ph  /\  A. x E. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.35-1 1603 Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)
 |-  ( E. x (
 ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
 
Theorem19.35i 1604 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  E. x ps )
 
Theorem19.25 1605 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
 
Theorem19.30dc 1606 Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  (DECID 
 E. x ps  ->  (
 A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) ) )
 
Theorem19.43 1607 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.33b2 1608 The antecedent provides a condition implying the converse of 19.33 1460. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1609 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
 |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph 
 \/  A. x ps )
 ) )
 
Theorem19.33bdc 1609 Converse of 19.33 1460 given  -.  ( E. x ph  /\ 
E. x ps ) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1608 (Contributed by Jim Kingdon, 23-Apr-2018.)
 |-  (DECID 
 E. x ph  ->  ( -.  ( E. x ph 
 /\  E. x ps )  ->  ( A. x (
 ph  \/  ps )  <->  (
 A. x ph  \/  A. x ps ) ) ) )
 
Theorem19.40 1610 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  ->  ( E. x ph  /\ 
 E. x ps )
 )
 
Theorem19.40-2 1611 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ph  /\  ps )  ->  ( E. x E. y ph  /\  E. x E. y ps )
 )
 
Theoremexintrbi 1612 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
 ps ) ) )
 
Theoremexintr 1613 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ( ph  /\  ps )
 ) )
 
Theoremalsyl 1614 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ch ) )  ->  A. x ( ph  ->  ch ) )
 
Theoremhbex 1615 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theoremnfex 1616 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |- 
 F/ x E. y ph
 
Theorem19.2 1617 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
 |-  ( A. x ph  ->  E. y ph )
 
Theoremi19.24 1618 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1603, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
 |-  ( ( A. x ph 
 ->  E. x ps )  ->  E. x ( ph  ->  ps ) )   =>    |-  ( ( A. x ph  ->  A. x ps )  ->  E. x ( ph  ->  ps ) )
 
Theoremi19.39 1619 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1603, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
 |-  ( ( A. x ph 
 ->  E. x ps )  ->  E. x ( ph  ->  ps ) )   =>    |-  ( ( E. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps )
 )
 
Theorem19.9ht 1620 A closed version of one direction of 19.9 1623. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( E. x ph  -> 
 ph ) )
 
Theorem19.9t 1621 A closed version of 19.9 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
 |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
 
Theorem19.9h 1622 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9 1623 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |-  ( E. x ph  <->  ph )
 
Theoremalexim 1624 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1598. (Contributed by Jim Kingdon, 2-Jul-2018.)
 |-  ( A. x ph  ->  -.  E. x  -.  ph )
 
Theoremexnalim 1625 One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
 |-  ( E. x  -.  ph 
 ->  -.  A. x ph )
 
Theoremexanaliim 1626 A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
 |-  ( E. x (
 ph  /\  -.  ps )  ->  -.  A. x (
 ph  ->  ps ) )
 
Theoremalexnim 1627 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( A. x E. y  -.  ph  ->  -.  E. x A. y ph )
 
Theoremax6blem 1628 If  x is not free in  ph, it is not free in  -.  ph. This theorem doesn't use ax6b 1629 compared to hbnt 1631. (Contributed by GD, 27-Jan-2018.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremax6b 1629 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

(Contributed by GD, 27-Jan-2018.)

 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1 1630  x is not free in  -.  A. x ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbnt 1631 Closed theorem version of bound-variable hypothesis builder hbn 1632. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph )
 )
 
Theoremhbn 1632 If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremhbnd 1633 Deduction form of bound-variable hypothesis builder hbn 1632. (Contributed by NM, 3-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps ) )
 
Theoremnfnt 1634 If  x is not free in  ph, then it is not free in  -.  ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
 |-  ( F/ x ph  ->  F/ x  -.  ph )
 
Theoremnfnd 1635 Deduction associated with nfnt 1634. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x  -.  ps )
 
Theoremnfn 1636 Inference associated with nfnt 1634. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x  -.  ph
 
Theoremnfdc 1637 If  x is not free in  ph, it is not free in DECID  ph. (Contributed by Jim Kingdon, 11-Mar-2018.)
 |- 
 F/ x ph   =>    |- 
 F/ xDECID 
 ph
 
Theoremmodal-5 1638 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( -.  A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
 
Theorem19.9d 1639 A deduction version of one direction of 19.9 1623. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( ps  ->  F/ x ph )   =>    |-  ( ps  ->  ( E. x ph  ->  ph )
 )
 
Theorem19.9hd 1640 A deduction version of one direction of 19.9 1623. This is an older variation of this theorem; new proofs should use 19.9d 1639. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ps  ->  (
 ph  ->  A. x ph )
 )   =>    |-  ( ps  ->  ( E. x ph  ->  ph )
 )
 
Theoremexcomim 1641 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
Theoremexcom 1642 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theorem19.12 1643 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theorem19.19 1644 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 E. x ps )
 )
 
Theorem19.21-2 1645 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
 |- 
 F/ x ph   &    |-  F/ y ph   =>    |-  ( A. x A. y (
 ph  ->  ps )  <->  ( ph  ->  A. x A. y ps ) )
 
Theoremnf2 1646 An alternate definition of df-nf 1437, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
 
Theoremnf3 1647 An alternate definition of df-nf 1437. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  A. x ( E. x ph 
 ->  ph ) )
 
Theoremnf4dc 1648 Variable  x is effectively not free in  ph iff  ph is always true or always false, given a decidability condition. The reverse direction, nf4r 1649, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  (DECID 
 E. x ph  ->  ( F/ x ph  <->  ( A. x ph 
 \/  A. x  -.  ph ) ) )
 
Theoremnf4r 1649 If  ph is always true or always false, then variable 
x is effectively not free in 
ph. The converse holds given a decidability condition, as seen at nf4dc 1648. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( ( A. x ph 
 \/  A. x  -.  ph )  ->  F/ x ph )
 
Theorem19.36i 1650 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |- 
 F/ x ps   &    |-  E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem19.36-1 1651 Closed form of 19.36i 1650. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  ps ) )
 
Theorem19.37-1 1652 One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  ->  ps )  ->  ( ph  ->  E. x ps )
 )
 
Theorem19.37aiv 1653* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.38 1654 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( E. x ph 
 ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
 
Theorem19.23t 1655 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |-  ( F/ x ps  ->  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) ) )
 
Theorem19.23 1656 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )
 
Theorem19.32dc 1657 Theorem 19.32 of [Margaris] p. 90, where  ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
 |- 
 F/ x ph   =>    |-  (DECID 
 ph  ->  ( A. x ( ph  \/  ps )  <->  (
 ph  \/  A. x ps ) ) )
 
Theorem19.32r 1658 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if  ph is decidable, as seen at 19.32dc 1657. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |- 
 F/ x ph   =>    |-  ( ( ph  \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.31r 1659 One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |- 
 F/ x ps   =>    |-  ( ( A. x ph  \/  ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.44 1660 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  ps ) )
 
Theorem19.45 1661 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  \/  ps )  <->  (
 ph  \/  E. x ps ) )
 
Theorem19.34 1662 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
 
Theorem19.41h 1663 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1664 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( E. x (
 ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.41 1664 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.42h 1665 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1666 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theorem19.42 1666 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theoremexcom13 1667 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
 
Theoremexrot3 1668 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4 1669 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ph  <->  E. z E. w E. x E. y ph )
 
Theoremnexr 1670 Inference from 19.8a 1569. (Contributed by Jeff Hankins, 26-Jul-2009.)
 |- 
 -.  E. x ph   =>    |- 
 -.  ph
 
Theoremexan 1671 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ph  /\ 
 ps )   =>    |- 
 E. x ( ph  /\ 
 ps )
 
Theoremhbexd 1672 Deduction form of bound-variable hypothesis builder hbex 1615. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( E. y ps  ->  A. x E. y ps ) )
 
Theoremeeor 1673 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  \/  ps )  <->  ( E. x ph  \/  E. y ps ) )
 
1.3.8  Equality theorems without distinct variables
 
Theorema9e 1674 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1423 through ax-14 1492 and ax-17 1506, all axioms other than ax-9 1511 are believed to be theorems of free logic, although the system without ax-9 1511 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
 |- 
 E. x  x  =  y
 
Theorema9ev 1675* At least one individual exists. Weaker version of a9e 1674. (Contributed by NM, 3-Aug-2017.)
 |- 
 E. x  x  =  y
 
Theoremax9o 1676 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Theoremequid 1677 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms.

This proof is similar to Tarski's and makes use of a dummy variable  y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

 |-  x  =  x
 
Theoremnfequid 1678 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
 |- 
 F/ y  x  =  x
 
Theoremstdpc6 1679 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1743.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
 |- 
 A. x  x  =  x
 
Theoremequcomi 1680 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax6evr 1681* A commuted form of a9ev 1675. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)
 |- 
 E. x  y  =  x
 
Theoremequcom 1682 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequcomd 1683 Deduction form of equcom 1682, symmetry of equality. For the versions for classes, see eqcom 2141 and eqcomd 2145. (Contributed by BJ, 6-Oct-2019.)
 |-  ( ph  ->  x  =  y )   =>    |-  ( ph  ->  y  =  x )
 
Theoremequcoms 1684 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ph )   =>    |-  ( y  =  x 
 ->  ph )
 
Theoremequtr 1685 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 1686 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  =  x 
 ->  z  =  y
 ) )
 
Theoremequtr2 1687 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
 
Theoremequequ1 1688 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  =  z  <-> 
 y  =  z ) )
 
Theoremequequ2 1689 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  =  x  <-> 
 z  =  y ) )
 
Theoremelequ1 1690 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  <->  y  e.  z ) )
 
Theoremelequ2 1691 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  <->  z  e.  y ) )
 
Theoremax11i 1692 Inference that has ax-11 1484 (without  A. y) as its conclusion and doesn't require ax-10 1483, ax-11 1484, or ax-12 1489 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
1.3.9  Axioms ax-10 and ax-11
 
Theoremax10o 1693 Show that ax-10o 1694 can be derived from ax-10 1483. An open problem is whether this theorem can be derived from ax-10 1483 and the others when ax-11 1484 is replaced with ax-11o 1795. See theorem ax10 1695 for the rederivation of ax-10 1483 from ax10o 1693.

Normally, ax10o 1693 should be used rather than ax-10o 1694, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Axiomax-10o 1694 Axiom ax-10o 1694 ("o" for "old") was the original version of ax-10 1483, before it was discovered (in May 2008) that the shorter ax-10 1483 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by theorem ax10o 1693.

Normally, ax10o 1693 should be used rather than ax-10o 1694, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremax10 1695 Rederivation of ax-10 1483 from original version ax-10o 1694. See theorem ax10o 1693 for the derivation of ax-10o 1694 from ax-10 1483.

This theorem should not be referenced in any proof. Instead, use ax-10 1483 above so that uses of ax-10 1483 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremhbae 1696 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfae 1697 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z A. x  x  =  y
 
Theoremhbaes 1698 Rule that applies hbae 1696 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z A. x  x  =  y  -> 
 ph )   =>    |-  ( A. x  x  =  y  ->  ph )
 
Theoremhbnae 1699 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremnfnae 1700 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z  -.  A. x  x  =  y
    < 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 10501-10600 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 13201-13283
  Copyright terms: Public domain < Previous  Next >