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Theorem ax4 32219
Description: Rederivation of axiom ax-4 1678 from ax-c4 32209 and other older axioms. See axc4 1910 for the derivation of ax-c4 32209 from ax-4 1678. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax4  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )

Proof of Theorem ax4
StepHypRef Expression
1 ax-c4 32209 . . 3  |-  ( A. x ( A. x
( ph  ->  ps )  ->  ( A. x ph  ->  ps ) )  -> 
( A. x (
ph  ->  ps )  ->  A. x ( A. x ph  ->  ps ) ) )
2 ax-c5 32208 . . . 4  |-  ( A. x ph  ->  ph )
3 ax-c5 32208 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
42, 3syl5 33 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  ps )
)
51, 4mpg 1667 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) )
6 ax-c4 32209 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
75, 6syl 17 1  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1665  ax-c5 32208  ax-c4 32209
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator