Theorem equsexh 2295

Description: An equivalence related to implicit substitution. See equsexhv 2124 for a version with a dv condition which does not require ax-13 2246. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
equsalh.1	|- ( ps -> A. x ps )
equsalh.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsalh.1	. . 3 |- ( ps -> A. x ps )
2	1	nf5i 2024	. 2 |- F/ x ps
3		equsalh.2	. 2 |- ( x = y -> ( ph <-> ps ) )
4	2, 3	equsex 2292	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)