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Theorem axc711 2221
Description: Proof of a single axiom that can replace both ax-c7 2193 and ax-11 1781. See axc711toc7 2223 and axc711to11 2224 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc711  |-  ( -. 
A. x  -.  A. y A. x ph  ->  A. y ph )

Proof of Theorem axc711
StepHypRef Expression
1 ax-11 1781 . . . . 5  |-  ( A. y A. x ph  ->  A. x A. y ph )
21con3i 135 . . . 4  |-  ( -. 
A. x A. y ph  ->  -.  A. y A. x ph )
32alimi 1605 . . 3  |-  ( A. x  -.  A. x A. y ph  ->  A. x  -.  A. y A. x ph )
43con3i 135 . 2  |-  ( -. 
A. x  -.  A. y A. x ph  ->  -. 
A. x  -.  A. x A. y ph )
5 ax-c7 2193 . 2  |-  ( -. 
A. x  -.  A. x A. y ph  ->  A. y ph )
64, 5syl 16 1  |-  ( -. 
A. x  -.  A. y A. x ph  ->  A. y ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-11 1781  ax-c7 2193
This theorem is referenced by:  axc711toc7  2223  axc711to11  2224
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