Theorem axc4 1865

Description: Show that the original axiom ax-c4 2217 can be derived from ax-4 1636 and others. See ax4 2227 for the rederivation of ax-4 1636 from ax-c4 2217.

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 1864	. . . 4 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 120	. . 3 |- ( A. x ph -> -. A. x -. A. x ph )
3		hbn1 1843	. . 3 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		hbn1 1843	. . . . 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 1638	. . 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 1637	. 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 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859

This theorem depends on definitions:  df-bi 185  df-ex 1618

This theorem is referenced by:  axc5c4c711  31549