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Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalimd 1501 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 1502 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 1503 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 1504 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 1505 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 1506* 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 1507* ax-17 1506 with antecedent. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremnfv 1508* 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 1509* nfv 1508 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1564. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
1.3.4  Introduce Axiom of Existence
 
Axiomax-i9 1510 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 1487 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 1674, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |- 
 E. x  x  =  y
 
Theoremax-9 1511 Derive ax-9 1511 from ax-i9 1510, the modified version for intuitionistic logic. Although ax-9 1511 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1510. (Contributed by NM, 3-Feb-2015.)
 |- 
 -.  A. x  -.  x  =  y
 
Theoremequidqe 1512 equid 1677 with some quantification and negation without using ax-4 1487 or ax-17 1506. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
 |- 
 -.  A. y  -.  x  =  x
 
Theoremax4sp1 1513 A special case of ax-4 1487 without using ax-4 1487 or ax-17 1506. (Contributed by NM, 13-Jan-2011.)
 |-  ( A. y  -.  x  =  x  ->  -.  x  =  x )
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1514  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 1515 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 1516 Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1517 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspsd 1518 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremnfbidf 1519 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 1520  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 1521  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x A. x ph
 
Theorema5i 1522 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremnfnf1 1523  x is not free in  F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/ x ph
 
Theoremhbim 1524 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 1525 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 1526 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 1527 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 1528 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 1529 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 1530 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 1531 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 1532 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 1533 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 1534 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 1535 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 1536 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 1562 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 1537 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1538 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.27h 1539 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 1540 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 1541 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 1542 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 1543 A closed form of nfan 1544. (Contributed by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  /\  ps )
 
Theoremnfan 1544 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 1545 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 1546 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 1547 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 1548 Deduction form of bound-variable hypothesis builder nf3an 1545. (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 1549 A closed form of hbim 1524. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremnfim1 1550 A closed form of nfim 1551. (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 1551 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 1552 Deduction form of bound-variable hypothesis builder hbim 1524. (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 1553 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 1554 Deduction form of bound-variable hypothesis builder hbbi 1527. (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 1555 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x A. y ph
 
Theoremnfnf 1556 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 1557 Closed form of nfal 1555. (Contributed by Jim Kingdon, 11-May-2018.)
 |-  ( A. y F/ x ph  ->  F/ x A. y ph )
 
Theoremnfa2 1558 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x A. y A. x ph
 
Theoremnfia1 1559 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ( A. x ph  ->  A. x ps )
 
Theorem19.21ht 1560 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 1561 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 1562 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 1563 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 1678. 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 1564 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 1565 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 1566 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 1567 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 1568 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 1569 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 1570 If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1569. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.23bi 1571 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremexlimih 1572 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 1573 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 1574 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1575 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 1575 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 1576 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 1577* 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 1577 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 1578 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 1579 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1580 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 1581 Inference associated with eximi 1579. (Contributed by BJ, 3-Feb-2018.)
 |- 
 E. x ph   &    |-  ( ph  ->  ps )   =>    |- 
 E. x ps
 
Theoremalinexa 1582 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremexbi 1583 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1584 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 1585 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 1586 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 1587 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdd 1588 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 1589 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 1590 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 1591 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 1592 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremexbidh 1593 Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremalbid 1594 Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbid 1595 Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremexsimpl 1596 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ph )
 
Theoremexsimpr 1597 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ps )
 
Theoremalexdc 1598 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1624. (Contributed by Jim Kingdon, 2-Jun-2018.)
 |-  ( A. xDECID  ph  ->  (
 A. x ph  <->  -.  E. x  -.  ph ) )
 
Theorem19.29 1599 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( A. x ph 
 /\  E. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r 1600 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |-  ( ( E. x ph 
 /\  A. x ps )  ->  E. x ( ph  /\ 
 ps ) )
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