Theorem axc4 1809

Description: Show that the original axiom ax-c4 2208 can be derived from ax-4 1612 and others. See ax4 2218 for the rederivation of ax-4 1612 from ax-c4 2208.

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 1808	. . . 4 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 120	. . 3 |- ( A. x ph -> -. A. x -. A. x ph )
3		hbn1 1787	. . 3 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		hbn1 1787	. . . . 5 |- ( -. A. x ph -> A. x -. A. x ph )
5	4	con1i 129	. . . 4 |- ( -. A. x -. A. x ph -> A. x ph )
6	5	alimi 1614	. . 3 |- ( A. x -. A. x -. A. x ph -> A. x A. x ph )
7	2, 3, 6	3syl 20	. 2 |- ( A. x ph -> A. x A. x ph )
8		alim 1613	. 2 |- ( A. x ( A. x ph -> ps ) -> ( A. x A. x ph -> A. x ps ) )
9	7, 8	syl5 32	1 |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803

This theorem depends on definitions:  df-bi 185  df-ex 1597

This theorem is referenced by:  axc5c4c711  30886