Theorem a4im 1159

Description: Specialization with implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The a4im 1159 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 992	. 2 |- (A.xph -> A.x(x = y -> A.xps))
5		ax-9o 1123	. 2 |- (A.x(x = y -> A.xps) -> ps)
6	4, 5	syl 10	1 |- (A.xph -> ps)

Syntax hints:   -> wi 3  A.wal 954   = wceq 956

This theorem is referenced by:  a4ime 1160  chvar 1167  a4imv 1207

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

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