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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfi 1601 Deduce that  x is not free in  ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  A. x ph )   =>    |- 
 F/ x ph
 
Theoremhbth 1602 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form  |-  ( ph  ->  A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in  ph." (Contributed by NM, 11-May-1993.)

 |-  ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfth 1603 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- 
 F/ x ph
 
Theoremnftru 1604 The true constant has no free variables. (This can also be proven in one step with nfv 1678, but this proof does not use ax-5 1675.) (Contributed by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x T.
 
Theoremnex 1605 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
 |- 
 -.  ph   =>    |- 
 -.  E. x ph
 
Theoremnfnth 1606 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
 |- 
 -.  ph   =>    |- 
 F/ x ph
 
1.4.3  Axiom scheme ax-4 (Quantified Implication)
 
Axiomax-4 1607 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1608 for labelling consistency. It should be used only by alim 1608. (Contributed by NM, 21-May-2008.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalim 1608 Restatement of Axiom ax-4 1607, for labelling consistency. It should be the only theorem using ax-4 1607. (Contributed by NM, 10-Jan-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalimi 1609 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorem2alimi 1610 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x A. y ph  ->  A. x A. y ps )
 
Theoremal2imi 1611 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( A. x ps  ->  A. x ch )
 )
 
Theoremalanimi 1612 Variant of al2imi 1611 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x ph 
 /\  A. x ps )  ->  A. x ch )
 
Theoremalimdh 1613 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalbi 1614 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( A. x ph  <->  A. x ps )
 )
 
Theoremalbii 1615 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x ph  <->  A. x ps )
 
Theorem2albii 1616 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x A. y ph  <->  A. x A. y ps )
 
Theoremalrimih 1617 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 9-Jan-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremhbxfrbi 1618 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2584 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfbii 1619 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   =>    |-  ( F/ x ph  <->  F/ x ps )
 
Theoremnfxfr 1620 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   &    |-  F/ x ps   =>    |-  F/ x ph
 
Theoremnfxfrd 1621 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  F/ x ps )   =>    |-  ( ch  ->  F/ x ph )
 
Theoremalex 1622 Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x ph  <->  -.  E. x  -.  ph )
 
Theoremexnal 1623 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x  -.  ph  <->  -. 
 A. x ph )
 
Theorem2nalexn 1624 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  A. x A. y ph  <->  E. x E. y  -.  ph )
 
Theorem2exnaln 1625 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ph  <->  -.  A. x A. y  -.  ph )
 
Theorem2nexaln 1626 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  <->  A. x A. y  -.  ph )
 
Theoremaleximi 1627 A variant of al2imi 1611: instead of applying  A. x quantifiers to the final implication, replace them with  E. x. A shorter proof is possible using nfa1 1840, sps 1809 and eximd 1825, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( E. x ps  ->  E. x ch )
 )
 
Theoremexim 1628 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
TheoremeximOLD 1629 Obsolete proof of exim 1628 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
Theoremeximi 1630 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1631 Inference adding two 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 1632 Inference associated with eximi 1630. (Contributed by BJ, 3-Feb-2018.)
 |- 
 E. x ph   &    |-  ( ph  ->  ps )   =>    |- 
 E. x ps
 
TheoremaleximiOLD 1633 Obsolete proof of aleximi 1627 as of 4-Sep-2019. (Contributed by Wolf Lammen, 18-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( E. x ps  ->  E. x ch )
 )
 
Theorem19.38 1634 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 1847 and 19.23t 1851. (Revised by Wolf Lammen, 2-Jan-2018.)
 |-  ( ( E. x ph 
 ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
 
Theoremalinexa 1635 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremalexn 1636 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x E. y  -.  ph  <->  -.  E. x A. y ph )
 
Theorem2exnexn 1637 Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
 |-  ( E. x A. y ph  <->  -.  A. x E. y  -.  ph )
 
Theoremexbi 1638 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1639 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 1640 Inference adding two 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 1641 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 )
 
Theoremexanali 1642 A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
 |-  ( E. x (
 ph  /\  -.  ps )  <->  -. 
 A. x ( ph  ->  ps ) )
 
Theoremexancom 1643 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdh 1644 Deduction from Theorem 19.21 of [Margaris] p. 90, see 19.21 1848. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximdh 1645 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 ) )
 
Theoremnexdh 1646 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbidh 1647 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidh 1648 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremexsimpl 1649 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 1650 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 )
 
Theorem19.40 1651 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
 |-  ( E. x (
 ph  /\  ps )  ->  ( E. x ph  /\ 
 E. x ps )
 )
 
Theorem19.26 1652 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. x ps ) )
 
Theorem19.26-2 1653 Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  /\  ps ) 
 <->  ( A. x A. y ph  /\  A. x A. y ps ) )
 
Theorem19.26-3an 1654 Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x (
 ph  /\  ps  /\  ch ) 
 <->  ( A. x ph  /\ 
 A. x ps  /\  A. x ch ) )
 
Theorem19.29 1655 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( A. x ph 
 /\  E. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r 1656 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 ) )
 
Theorem19.29r2 1657 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 1658 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 1659 Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35OLD 1660 Obsolete proof of 19.35 1659 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35i 1661 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 21-Jun-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  E. x ps )
 
Theorem19.35ri 1662 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x ph  ->  E. x ps )   =>    |-  E. x ( ph  ->  ps )
 
Theorem19.25 1663 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
 
Theorem19.30 1664 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  \/  ps )  ->  ( A. x ph  \/  E. x ps )
 )
 
Theorem19.43 1665 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.43OLD 1666 Obsolete proof of 19.43 1665 as of 3-May-2017. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.33 1667 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( ( A. x ph 
 \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.33b 1668 The antecedent provides a condition implying the converse of 19.33 1667. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
 |-  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph 
 \/  A. x ps )
 ) )
 
Theorem19.40-2 1669 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 )
 )
 
Theoremalbiim 1670 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ph ) ) )
 
Theorem2albiim 1671 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  <->  ps )  <->  ( A. x A. y ( ph  ->  ps )  /\  A. x A. y ( ps  ->  ph ) ) )
 
Theoremexintrbi 1672 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 1673 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 1674 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 ) )
 
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d
 
Axiomax-5 1675* Axiom of Distinctness. This axiom quantifies 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.

(See comments in ax5ALT 2224 about the logical redundancy of ax-5 1675 in the presence of our obsolete axioms.)

This axiom essentially says that if  x does not occur in  ph, i.e.  ph does not depend on  x in any way, then we can add the quantifier  A. x to  ph with no further assumptions. By sp 1803, we can also remove the quantifier (unconditionally). (Contributed by NM, 10-Jan-1993.)

 |-  ( ph  ->  A. x ph )
 
Theoremax5d 1676* ax-5 1675 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremax5e 1677* A rephrasing of ax-5 1675 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
 |-  ( E. x ph  -> 
 ph )
 
Theoremnfv 1678* 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 1679* nfv 1678 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1859. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
Theoremalimdv 1680* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  A. x ch ) )
 
Theoremeximdv 1681* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theorem2alimdv 1682* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps  ->  A. x A. y ch ) )
 
Theorem2eximdv 1683* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  E. x E. y ch ) )
 
Theoremalbidv 1684* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidv 1685* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theorem2albidv 1686* Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps 
 <-> 
 A. x A. y ch ) )
 
Theorem2exbidv 1687* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps 
 <-> 
 E. x E. y ch ) )
 
Theorem3exbidv 1688* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
 
Theorem4exbidv 1689* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z E. w ps  <->  E. x E. y E. z E. w ch ) )
 
Theoremalrimiv 1690* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 21-Jun-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalrimivv 1691* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x A. y ps )
 
Theoremalrimdv 1692* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch ) )
 
Theoremexlimiv 1693* Inference from Theorem 19.23 of [Margaris] p. 90, see 19.23 1852.

This inference, along with our many variants such as rexlimdv 2948, 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.appstate.edu/~hirstjl/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 (fictitious) 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 1693 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, 21-Jun-1993.) Remove dependencies on ax-6 1714 and ax-8 1764. (Revised by Wolf Lammen, 4-Dec-2017.)

 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimivv 1694* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  ps )
 
Theoremexlimdv 1695* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1714, ax-7 1734. (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch ) )
 
Theoremexlimdvv 1696* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  ch ) )
 
Theoremexlimddv 1697* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremnfdv 1698* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )   =>    |-  ( ph  ->  F/ x ps )
 
Theorem2ax5 1699* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
 |-  ( ph  ->  A. x A. y ph )
 
1.4.5  Equality predicate (continued)

The equality predicate was introduced above in wceq 1374 for use by df-tru 1377. See the comments in that section. In this section, we continue with the first "real" use of it.

 
Theoremweq 1700 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1700 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 1374. This lets us avoid overloading the  = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1700 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1374. 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
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