Theorem a4a 1142

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

Hypotheses

Ref	Expression
a4a.1	|- (ps -> A.xps)
a4a.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
a4a	|- (A.xph -> ps)

Proof of Theorem a4a

Step	Hyp	Ref	Expression
1		a4a.2	. . . 4 |- (x = y -> (ph -> ps))
2		a4a.1	. . . 4 |- (ps -> A.xps)
3	1, 2	syl6com 53	. . 3 |- (ph -> (x = y -> A.xps))
4	3	19.20i 968	. 2 |- (A.xph -> A.x(x = y -> A.xps))
5		ax9 1110	. 2 |- (A.x(x = y -> A.xps) -> ps)
6	4, 5	syl 10	1 |- (A.xph -> ps)

Syntax hints:   -> wi 3  A.wal 950   = wceq 1099

This theorem is referenced by:  a4c 1143  chvar 1150  a4b 1191

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-9 1102

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

Copyright terms: Public domain