Theorem a4at 1141

Description: Closed theorem form of a4a 1142.

Assertion

Ref	Expression
a4at	|- (A.x((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (A.xph -> ps))

Proof of Theorem a4at

Step	Hyp	Ref	Expression
1		imim2 14	. . . . . 6 |- ((ps -> A.xps) -> ((ph -> ps) -> (ph -> A.xps)))
2	1	imim2d 25	. . . . 5 |- ((ps -> A.xps) -> ((x = y -> (ph -> ps)) -> (x = y -> (ph -> A.xps))))
3	2	imp 350	. . . 4 |- (((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (x = y -> (ph -> A.xps)))
4	3	com23 32	. . 3 |- (((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (ph -> (x = y -> A.xps)))
5	4	19.20ii 971	. 2 |- (A.x((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (A.xph -> A.x(x = y -> A.xps)))
6		ax9 1110	. 2 |- (A.x(x = y -> A.xps) -> ps)
7	5, 6	syl6 22	1 |- (A.x((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (A.xph -> ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950   = wceq 1099

This theorem is referenced by:  a12lem1 1353

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

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