Theorem axc4 1910

Description: Show that the original axiom ax-c4 32165 can be derived from ax-4 1678 and others. See ax4 32175 for the rederivation of ax-4 1678 from ax-c4 32165.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc4	|- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )

Proof of Theorem axc4

Step	Hyp	Ref	Expression
1		sp 1909	. . . 4 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 123	. . 3 |- ( A. x ph -> -. A. x -. A. x ph )
3		hbn1 1887	. . 3 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		hbn1 1887	. . . . 5 |- ( -. A. x ph -> A. x -. A. x ph )
5	4	con1i 132	. . . 4 |- ( -. A. x -. A. x ph -> A. x ph )
6	5	alimi 1680	. . 3 |- ( A. x -. A. x -. A. x ph -> A. x A. x ph )
7	2, 3, 6	3syl 18	. 2 |- ( A. x ph -> A. x A. x ph )
8		alim 1679	. 2 |- ( A. x ( A. x ph -> ps ) -> ( A. x A. x ph -> A. x ps ) )
9	7, 8	syl5 33	1 |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904

This theorem depends on definitions:  df-bi 188  df-ex 1660

This theorem is referenced by:  axc5c4c711  36393