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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
StepHypRef Expression
1 ax-c5 32367 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 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 )
54con1i 132 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1678 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 18 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff setvar class
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
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