Theorem equsex 2143

Description: An equivalence related to implicit substitution. See equsexv 2085 for a version with a dv condition which does not require ax-13 2104. See equsexALT 2144 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)

Hypotheses

Ref	Expression
equsex.nf	|- F/ x ps
equsex.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
equsex	|- ( E. x ( x = y /\ ph ) <-> ps )

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		equsex.nf	. . 3 |- F/ x ps
2		equsex.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimpa 492	. . 3 |- ( ( x = y /\ ph ) -> ps )
4	1, 3	exlimi 2015	. 2 |- ( E. x ( x = y /\ ph ) -> ps )
5	1, 2	equsal 2141	. . 3 |- ( A. x ( x = y -> ph ) <-> ps )
6		equs4 2140	. . 3 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7	5, 6	sylbir 218	. 2 |- ( ps -> E. x ( x = y /\ ph ) )
8	4, 7	impbii 192	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   A.wal 1450   E.wex 1671   F/wnf 1675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676

This theorem is referenced by:  equsexh  2145  sb5rf  2271