Theorem ax4 32466

Description: Rederivation of axiom ax-4 1682 from ax-c4 32456 and other older axioms. See axc4 1938 for the derivation of ax-c4 32456 from ax-4 1682. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax4	|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )

Proof of Theorem ax4

Step	Hyp	Ref	Expression
1		ax-c4 32456	. . 3 |- ( A. x ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) ) -> ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) ) )
2		ax-c5 32455	. . . 4 |- ( A. x ph -> ph )
3		ax-c5 32455	. . . 4 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
4	2, 3	syl5 33	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) )
5	1, 4	mpg 1671	. 2 |- ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) )
6		ax-c4 32456	. 2 |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )
7	5, 6	syl 17	1 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )

Syntax hints:   -> wi 4   A.wal 1442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1669  ax-c5 32455  ax-c4 32456

This theorem is referenced by: (None)