Theorem hba1-o 2207

Description: x is not free in A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
hba1-o	|- ( A. x ph -> A. x A. x ph )

Proof of Theorem hba1-o

Step	Hyp	Ref	Expression
1		ax-4 2193	. . 3 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 114	. 2 |- ( A. x ph -> -. A. x -. A. x ph )
3		ax6 2205	. 2 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		ax6 2205	. . . 4 |- ( -. A. x ph -> A. x -. A. x ph )
5	4	con1i 123	. . 3 |- ( -. A. x -. A. x ph -> A. x ph )
6	5	alimi 1565	. 2 |- ( A. x -. A. x -. A. x ph -> A. x A. x ph )
7	2, 3, 6	3syl 19	1 |- ( A. x ph -> A. x A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1546

This theorem is referenced by:  a5i-o  2208  nfa1-o  2224  ax67to6  2225  ax467to6  2229  dvelimf-o  2238  ax11indalem  2255  ax11inda2ALT  2256  ax11inda  2258

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-4 2193  ax-5o 2194  ax-6o 2195