Theorem a4ime 1162

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 1006	. . . 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 1161	. . 3 |- (A.x -. ps -> -. ph)
6	5	con2i 97	. 2 |- (ph -> -. A.x -. ps)
7		df-ex 983	. 2 |- (E.xps <-> -. A.x -. ps)
8	6, 7	sylibr 200	1 |- (ph -> E.xps)

Syntax hints:  -. wn 2   -> wi 3  A.wal 956   = wceq 958  E.wex 982

This theorem is referenced by:  a4imed 1163  a4imev 1275  fine 10442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-ex 983

Copyright terms: Public domain