Theorem a4ime 1160

Description: Existential introduction with implicit substitution. Compare Lemma 14 of [Tarski] p. 70.

Hypotheses

Ref	Expression
a4ime.1	|- (ph -> A.xph)
a4ime.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
a4ime	|- (ph -> E.xps)

Proof of Theorem a4ime

Step	Hyp	Ref	Expression
1		a4ime.1	. . . . 5 |- (ph -> A.xph)
2	1	hbn 1004	. . . 4 |- (-. ph -> A.x -. ph)
3		a4ime.2	. . . . 5 |- (x = y -> (ph -> ps))
4	3	con3d 95	. . . 4 |- (x = y -> (-. ps -> -. ph))
5	2, 4	a4im 1159	. . 3 |- (A.x -. ps -> -. ph)
6	5	con2i 97	. 2 |- (ph -> -. A.x -. ps)
7		df-ex 981	. 2 |- (E.xps <-> -. A.x -. ps)
8	6, 7	sylibr 200	1 |- (ph -> E.xps)

Syntax hints:  -. wn 2   -> wi 3  A.wal 954   = wceq 956  E.wex 980

This theorem is referenced by:  a4imed 1161  a4imev 1273  fine 10449  fineOLD 10450

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-ex 981

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