Theorem ax4fromc4 34179

Description: Rederivation of axiom ax-4 1737 from ax-c4 34169, ax-c5 34168, ax-gen 1722 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2130 for the derivation of ax-c4 34169 from ax-4 1737. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ax4fromc4

Step	Hyp	Ref	Expression
1		ax-c4 34169	. . 3 |- ( A. x ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) ) -> ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) ) )
2		ax-c5 34168	. . . 4 |- ( A. x ph -> ph )
3		ax-c5 34168	. . . 4 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
4	2, 3	syl5 34	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) )
5	1, 4	mpg 1724	. 2 |- ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) )
6		ax-c4 34169	. 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 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1722  ax-c5 34168  ax-c4 34169

This theorem is referenced by: (None)