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Theorem hba1-o 2154
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
StepHypRef Expression
1 ax-4 2140 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 112 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 ax6 2152 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 ax6 2152 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 121 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1559 . 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 set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1540
This theorem is referenced by:  a5i-o  2155  nfa1-o  2171  ax67to6  2172  ax467to6  2176  dvelimf-o  2185  ax11indalem  2202  ax11inda2ALT  2203  ax11inda  2205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-4 2140  ax-5o 2141  ax-6o 2142
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