Theorem ax4fromc4

Description: Rederivation of Axiom ax-4 from ax-c4 , ax-c5 , ax-gen and minimal implicational calculus { ax-mp , ax-1 , ax-2 }. See axc4 for the derivation of ax-c4 from ax-4 . (Contributed by NM, 23-May-2008) (Proof modification is discouraged.) Use ax-4 instead. (New usage is discouraged.)

	Ref	Expression
Assertion	ax4fromc4	|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )

Proof

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