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Theorem hba1-o 2230
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
StepHypRef Expression
1 ax-c5 2216 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 120 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 ax10 2228 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 ax10 2228 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 129 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1638 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 20 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff setvar class
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-c5 2216  ax-c4 2217  ax-c7 2218
This theorem is referenced by:  axc4i-o  2231  nfa1-o  2247  axc711toc7  2248  axc5c711toc7  2252  dvelimf-o  2261  ax12indalem  2277  ax12inda2ALT  2278  ax12inda  2280
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