Theorem hba1-o 32381

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, 24-Jan-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-c5 32367	. . 3 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 123	. 2 |- ( A. x ph -> -. A. x -. A. x ph )
3		ax10 32379	. 2 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		ax10 32379	. . . 4 |- ( -. A. x ph -> A. x -. A. x ph )
5	4	con1i 132	. . 3 |- ( -. A. x -. A. x ph -> A. x ph )
6	5	alimi 1678	. 2 |- ( A. x -. A. x -. A. x ph -> A. x A. x ph )
7	2, 3, 6	3syl 18	1 |- ( A. x ph -> A. x A. x ph )

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 1663  ax-4 1676  ax-c5 32367  ax-c4 32368  ax-c7 32369

This theorem is referenced by:  axc4i-o  32382  nfa1-o  32398  axc711toc7  32399  axc5c711toc7  32403  dvelimf-o  32412  ax12indalem  32428  ax12inda2ALT  32429  ax12inda  32431