Theorem hba1-o 34182

Description: The setvar 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 34168	. . 3 |- ( A. x -. A. x ph -> -. A. x ph )
2	1	con2i 134	. 2 |- ( A. x ph -> -. A. x -. A. x ph )
3		ax10fromc7 34180	. 2 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
4		ax10fromc7 34180	. . . 4 |- ( -. A. x ph -> A. x -. A. x ph )
5	4	con1i 144	. . 3 |- ( -. A. x -. A. x ph -> A. x ph )
6	5	alimi 1739	. 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 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-c5 34168  ax-c4 34169  ax-c7 34170

This theorem is referenced by:  axc4i-o  34183  nfa1-o  34200  axc711toc7  34201  axc5c711toc7  34205  dvelimf-o  34214  ax12indalem  34230  ax12inda2ALT  34231  ax12inda  34233