Theorem a4eiv 1269

Description: Inference from existential specialization with implicit substitution.

Hypotheses

Ref	Expression
a4eiv.1	|- (x = y -> (ph <-> ps))
a4eiv.2	|- ps

Assertion

Ref	Expression
a4eiv	|- E.xph

Distinct variable group:   ps,x

Proof of Theorem a4eiv

Step	Hyp	Ref	Expression
1		a4eiv.2	. 2 |- ps
2		a4eiv.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	biimprd 154	. . 3 |- (x = y -> (ps -> ph))
4	3	a4imev 1268	. 2 |- (ps -> E.xph)
5	1, 4	ax-mp 7	1 |- E.xph

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953  E.wex 977

This theorem is referenced by:  uniiunlem 2122  elirrv 4570

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119

This theorem depends on definitions:  df-bi 147  df-ex 978

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