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Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxi12 1501 Proof that ax-i12 1494 follows from ax-bndl 1496. So that we can track which theorems rely on ax-bndl 1496, proofs should reference ax12or 1495 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremalequcom 1502 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremalequcoms 1503 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremnalequcoms 1504 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  ->  ph )
 
Theoremnfr 1505 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1506 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1507 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremalimd 1508 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalrimi 1509 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremnfd 1510 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremnfdh 1511 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremnfrimi 1512 Moving an antecedent outside  F/. (Contributed by Jim Kingdon, 23-Mar-2018.)
 |- 
 F/ x ph   &    |-  F/ x (
 ph  ->  ps )   =>    |-  ( ph  ->  F/ x ps )
 
1.3.3  Axiom ax-17 - first use of the $d distinct variable statement
 
Axiomax-17 1513* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(Contributed by NM, 5-Aug-1993.)

 |-  ( ph  ->  A. x ph )
 
Theorema17d 1514* ax-17 1513 with antecedent. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremnfv 1515* If  x is not present in  ph, then  x is not free in  ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph
 
Theoremnfvd 1516* nfv 1515 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1572. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
1.3.4  Introduce Axiom of Existence
 
Axiomax-i9 1517 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1497 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that  x and  y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1683, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |- 
 E. x  x  =  y
 
Theoremax-9 1518 Derive ax-9 1518 from ax-i9 1517, the modified version for intuitionistic logic. Although ax-9 1518 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1517. (Contributed by NM, 3-Feb-2015.)
 |- 
 -.  A. x  -.  x  =  y
 
Theoremequidqe 1519 equid 1688 with some quantification and negation without using ax-4 1497 or ax-17 1513. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
 |- 
 -.  A. y  -.  x  =  x
 
Theoremax4sp1 1520 A special case of ax-4 1497 without using ax-4 1497 or ax-17 1513. (Contributed by NM, 13-Jan-2011.)
 |-  ( A. y  -.  x  =  x  ->  -.  x  =  x )
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1521  x is not free in  A. x ph. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Axiomax-i5r 1522 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( A. x ph 
 ->  A. x ps )  ->  A. x ( A. x ph  ->  ps )
 )
 
1.3.6  Predicate calculus including ax-4, without distinct variables
 
Theoremspi 1523 Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1524 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspsd 1525 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremnfbidf 1526 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( F/ x ps  <->  F/ x ch )
 )
 
Theoremhba1 1527  x is not free in  A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremnfa1 1528  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x A. x ph
 
Theoremaxc4i 1529 Inference version of 19.21 1570. (Contributed by NM, 3-Jan-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorema5i 1530 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremnfnf1 1531  x is not free in  F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/ x ph
 
Theoremhbim 1532 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  ->  ps )  ->  A. x ( ph  ->  ps )
 )
 
Theoremhbor 1533 If  x is not free in  ph and  ps, it is not free in  ( ph  \/  ps ). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  \/  ps )  ->  A. x ( ph  \/  ps )
 )
 
Theoremhban 1534 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  /\ 
 ps )  ->  A. x ( ph  /\  ps )
 )
 
Theoremhbbi 1535 If  x is not free in  ph and  ps, it is not free in  ( ph  <->  ps ). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  <->  ps )  ->  A. x ( ph  <->  ps ) )
 
Theoremhb3or 1536 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  \/  ps  \/  ch ). (Contributed by NM, 14-Sep-2003.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  \/  ps  \/  ch )  ->  A. x (
 ph  \/  ps  \/  ch ) )
 
Theoremhb3an 1537 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by NM, 14-Sep-2003.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  A. x (
 ph  /\  ps  /\  ch ) )
 
Theoremhba2 1538 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
 |-  ( A. y A. x ph  ->  A. x A. y A. x ph )
 
Theoremhbia1 1539 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
 |-  ( ( A. x ph 
 ->  A. x ps )  ->  A. x ( A. x ph  ->  A. x ps ) )
 
Theorem19.3h 1540 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x ph  <->  ph )
 
Theorem19.3 1541 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x ph  <->  ph )
 
Theorem19.16 1542 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 A. x ps )
 )
 
Theorem19.17 1543 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  ps ) )
 
Theorem19.21h 1544 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph". New proofs should use 19.21 1570 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theorem19.21bi 1545 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1546 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.27h 1547 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.27 1548 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28h 1549 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.28 1550 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theoremnfan1 1551 A closed form of nfan 1552. (Contributed by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  /\  ps )
 
Theoremnfan 1552 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  /\  ps )
 
Theoremnf3an 1553 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  F/ x ch   =>    |- 
 F/ x ( ph  /\ 
 ps  /\  ch )
 
Theoremnford 1554 If in a context  x is not free in  ps and  ch, it is not free in  ( ps  \/  ch ). (Contributed by Jim Kingdon, 29-Oct-2019.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  \/  ch ) )
 
Theoremnfand 1555 If in a context  x is not free in  ps and  ch, it is not free in  ( ps  /\  ch ). (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch ) )
 
Theoremnf3and 1556 Deduction form of bound-variable hypothesis builder nf3an 1553. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  F/ x th )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch  /\  th ) )
 
Theoremhbim1 1557 A closed form of hbim 1532. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremnfim1 1558 A closed form of nfim 1559. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  ->  ps )
 
Theoremnfim 1559 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  ->  ps )
 
Theoremhbimd 1560 Deduction form of bound-variable hypothesis builder hbim 1532. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremnfor 1561 If  x is not free in  ph and  ps, it is not free in  ( ph  \/  ps ). (Contributed by Jim Kingdon, 11-Mar-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  \/  ps )
 
Theoremhbbid 1562 Deduction form of bound-variable hypothesis builder hbbi 1535. (Contributed by NM, 1-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   =>    |-  ( ph  ->  (
 ( ps  <->  ch )  ->  A. x ( ps  <->  ch ) ) )
 
Theoremnfal 1563 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1497. (Revised by Gino Giotto, 25-Aug-2024.)
 |- 
 F/ x ph   =>    |- 
 F/ x A. y ph
 
Theoremnfnf 1564 If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |- 
 F/ x F/ y ph
 
Theoremnfalt 1565 Closed form of nfal 1563. (Contributed by Jim Kingdon, 11-May-2018.)
 |-  ( A. y F/ x ph  ->  F/ x A. y ph )
 
Theoremnfa2 1566 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x A. y A. x ph
 
Theoremnfia1 1567 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ( A. x ph  ->  A. x ps )
 
Theorem19.21ht 1568 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theorem19.21t 1569 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
 |-  ( F/ x ph  ->  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theorem19.21 1570 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph". (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremstdpc5 1571 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis  F/ x ph can be thought of as emulating " x is not free in  ph". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example  x would not (for us) be free in  x  =  x by nfequid 1689. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
 
Theoremnfimd 1572 If in a context  x is not free in  ps and  ch, then it is not free in  ( ps  ->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  ->  ch ) )
 
Theoremaaanh 1573 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x A. y ( ph  /\  ps ) 
 <->  ( A. x ph  /\ 
 A. y ps )
 )
 
Theoremaaan 1574 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. y ps ) )
 
Theoremnfbid 1575 If in a context  x is not free in  ps and  ch, then it is not free in  ( ps  <->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  <->  ch ) )
 
Theoremnfbi 1576 If  x is not free in  ph and  ps, then it is not free in  ( ph  <->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  <->  ps )
 
1.3.7  The existential quantifier
 
Theorem19.8a 1577 If a wff is true, then it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  E. x ph )
 
Theorem19.8ad 1578 If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1577. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.23bi 1579 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremexlimih 1580 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimi 1581 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ps   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimd2 1582 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1583 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremexlimdh 1583 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremexlimd 1584 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremexlimiv 1585* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.

In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element  x exists satisfying a wff, i.e.  E. x ph ( x ) where  ph ( x ) has  x free, then we can use  ph ( C  ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original  ph (containing  x) as an antecedent for the main part of the proof. We eventually arrive at  ( ph  ->  ps ) where  ps is the theorem to be proved and does not contain  x. Then we apply exlimiv 1585 to arrive at  ( E. x ph  ->  ps ). Finally, we separately prove  E. x ph and detach it with modus ponens ax-mp 5 to arrive at the final theorem  ps. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)

 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexim 1586 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
Theoremeximi 1587 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1588 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  E. x E. y ps )
 
Theoremeximii 1589 Inference associated with eximi 1587. (Contributed by BJ, 3-Feb-2018.)
 |- 
 E. x ph   &    |-  ( ph  ->  ps )   =>    |- 
 E. x ps
 
Theoremalinexa 1590 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremexbi 1591 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1592 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x ph  <->  E. x ps )
 
Theorem2exbii 1593 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y ph  <->  E. x E. y ps )
 
Theorem3exbii 1594 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y E. z ph  <->  E. x E. y E. z ps )
 
Theoremexancom 1595 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdd 1596 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremalrimd 1597 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximdh 1598 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremeximd 1599 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremnexd 1600 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
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