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Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfexdc 1501 Defining  E. x ph given decidability. It is common in classical logic to define  E. x ph as  -.  A. x -.  ph but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1502. (Contributed by Jim Kingdon, 15-Mar-2018.)
 |-  (DECID 
 E. x ph  ->  ( E. x ph  <->  -.  A. x  -.  ph ) )
 
Theoremexalim 1502 One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1501. (Contributed by Jim Kingdon, 29-Jul-2018.)
 |-  ( E. x ph  ->  -.  A. x  -.  ph )
 
1.3.2  Equality predicate (continued)

The equality predicate was introduced above in wceq 1353 for use by df-tru 1356. See the comments in that section. In this section, we continue with the first "real" use of it.

 
Theoremweq 1503 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1503 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1353. This lets us avoid overloading the  = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1503 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1353. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

 wff  x  =  y
 
Axiomax-8 1504 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1709). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1504 through ax-16 1814 are the axioms having to do with equality, substitution, and logical properties of our binary predicate  e. (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1814 and ax-17 1526 are still valid even when  x,  y, and  z are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1814 and ax-17 1526 only. (Contributed by NM, 5-Aug-1993.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Axiomax-10 1505 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 1716 ("o" for "old") and was replaced with this shorter ax-10 1505 in May 2008. The old axiom is proved from this one as Theorem ax10o 1715. Conversely, this axiom is proved from ax-10o 1716 as Theorem ax10 1717. (Contributed by NM, 5-Aug-1993.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Axiomax-11 1506 Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent  A. x ( x  =  y  ->  ph ) is a way of expressing " y substituted for  x in wff  ph " (cf. sb6 1886). 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.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1827, ax11v2 1820 and ax-11o 1823. (Contributed by NM, 5-Aug-1993.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Axiomax-i12 1507 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever  z is distinct from  x and  y, and  x  =  y is true, then  x  =  y quantified with  z is also true. In other words,  z is irrelevant to the truth of 
x  =  y. Axiom scheme C9' 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.

This axiom has been modified from the original ax12 1512 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1508 instead, for labeling consistency. (New usage is discouraged.)

 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremax12or 1508 Alias for ax-i12 1507, to be used in place of it for labeling consistency. (Contributed by NM, 3-Feb-2015.)
 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y ) ) )
 
Axiomax-bndl 1509 Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as  A. z z  =  x  \/  -.  A. z z  =  x. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object.  A. z
z  =  x holds if  z and  x are the same variable, likewise for  z and  y, and  A. x A. z ( x  =  y  ->  A. z
x  =  y ) holds if  z is distinct from the others (and the universe has at least two objects).

As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability).

This axiom implies ax-i12 1507 as can be seen at axi12 1514. Whether ax-bndl 1509 can be proved from the remaining axioms including ax-i12 1507 is not known.

The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)

 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y ) ) )
 
Axiomax-4 1510 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all  x, it is true for any specific  x (that would typically occur as a free variable in the wff substituted for  ph). (A free variable is one that does not occur in the scope of a quantifier:  x and  y are both free in  x  =  y, but only  x is free in  A. y x  =  y.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1449. Conditional forms of the converse are given by ax12 1512, ax-16 1814, and ax-17 1526.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from  x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1775.

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

 |-  ( A. x ph  -> 
 ph )
 
Theoremsp 1511 Specialization. Another name for ax-4 1510. (Contributed by NM, 21-May-2008.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremax12 1512 Rederive the original version of the axiom from ax-i12 1507. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremhbequid 1513 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.

The proof uses only ax-8 1504 and ax-i12 1507 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1530, even though Theorem equid 1701 cannot. A shorter proof using ax-i9 1530 is obtainable from equid 1701 and hbth 1463. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)

 |-  ( x  =  x 
 ->  A. y  x  =  x )
 
Theoremaxi12 1514 Proof that ax-i12 1507 follows from ax-bndl 1509. So that we can track which theorems rely on ax-bndl 1509, proofs should reference ax12or 1508 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 1515 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 1516 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremnalequcoms 1517 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 1518 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1519 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1520 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 1521 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 1522 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 1523 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 1524 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 1525 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 1526* 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 1527* ax-17 1526 with antecedent. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremnfv 1528* 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 1529* nfv 1528 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1585. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
1.3.4  Introduce Axiom of Existence
 
Axiomax-i9 1530 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 1510 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 1696, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |- 
 E. x  x  =  y
 
Theoremax-9 1531 Derive ax-9 1531 from ax-i9 1530, the modified version for intuitionistic logic. Although ax-9 1531 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1530. (Contributed by NM, 3-Feb-2015.)
 |- 
 -.  A. x  -.  x  =  y
 
Theoremequidqe 1532 equid 1701 with some quantification and negation without using ax-4 1510 or ax-17 1526. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
 |- 
 -.  A. y  -.  x  =  x
 
Theoremax4sp1 1533 A special case of ax-4 1510 without using ax-4 1510 or ax-17 1526. (Contributed by NM, 13-Jan-2011.)
 |-  ( A. y  -.  x  =  x  ->  -.  x  =  x )
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1534  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 1535 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 1536 Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1537 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspsd 1538 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremnfbidf 1539 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 1540  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 1541  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x A. x ph
 
Theoremaxc4i 1542 Inference version of 19.21 1583. (Contributed by NM, 3-Jan-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorema5i 1543 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremnfnf1 1544  x is not free in  F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/ x ph
 
Theoremhbim 1545 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 1546 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 1547 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 1548 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 1549 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 1550 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 1551 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 1552 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 1553 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 1554 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 1555 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 1556 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 1557 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 1583 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 1558 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1559 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.27h 1560 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 1561 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 1562 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 1563 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 1564 A closed form of nfan 1565. (Contributed by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  /\  ps )
 
Theoremnfan 1565 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 1566 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 1567 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 1568 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 1569 Deduction form of bound-variable hypothesis builder nf3an 1566. (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 1570 A closed form of hbim 1545. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremnfim1 1571 A closed form of nfim 1572. (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 1572 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 1573 Deduction form of bound-variable hypothesis builder hbim 1545. (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 1574 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 1575 Deduction form of bound-variable hypothesis builder hbbi 1548. (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 1576 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 1510. (Revised by Gino Giotto, 25-Aug-2024.)
 |- 
 F/ x ph   =>    |- 
 F/ x A. y ph
 
Theoremnfnf 1577 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 1578 Closed form of nfal 1576. (Contributed by Jim Kingdon, 11-May-2018.)
 |-  ( A. y F/ x ph  ->  F/ x A. y ph )
 
Theoremnfa2 1579 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x A. y A. x ph
 
Theoremnfia1 1580 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ( A. x ph  ->  A. x ps )
 
Theorem19.21ht 1581 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 1582 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 1583 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 1584 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 1702. 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 1585 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 1586 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 1587 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 1588 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 1589 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 1590 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 1591 If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1590. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.23bi 1592 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremexlimih 1593 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 1594 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 1595 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1596 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 1596 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 1597 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 1598* 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 1598 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 1599 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 1600 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
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