Theorem a4im 1201

Description: Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The a4im 1201 series of theorems requires that only one direction of the substitution hypothesis hold.

Hypotheses

Ref	Expression
a4im.1	|- (ps -> A.xps)
a4im.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
a4im	|- (A.xph -> ps)

Proof of Theorem a4im

Step	Hyp	Ref	Expression
1		a4im.2	. . . 4 |- (x = y -> (ph -> ps))
2		a4im.1	. . . 4 |- (ps -> A.xps)
3	1, 2	syl6com 53	. . 3 |- (ph -> (x = y -> A.xps))
4	3	19.20i 1033	. 2 |- (A.xph -> A.x(x = y -> A.xps))
5		ax-9o 1164	. 2 |- (A.x(x = y -> A.xps) -> ps)
6	4, 5	syl 10	1 |- (A.xph -> ps)

Syntax hints:   -> wi 3  A.wal 995   = wceq 997

This theorem is referenced by:  a4ime 1202  chvar 1209  a4imv 1249

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-mp 7  ax-gen 1004  ax-4 1014  ax-5o 1016  ax-9o 1164

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