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Theorem ax5 2229
Description: Rederivation of axiom ax-5 1567 from ax-5o 2219 and other older axioms. See ax5o 1767 for the derivation of ax-5o 2219 from ax-5 1567. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax5  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )

Proof of Theorem ax5
StepHypRef Expression
1 ax-5o 2219 . . 3  |-  ( A. x ( A. x
( ph  ->  ps )  ->  ( A. x ph  ->  ps ) )  -> 
( A. x (
ph  ->  ps )  ->  A. x ( A. x ph  ->  ps ) ) )
2 ax-4 2218 . . . 4  |-  ( A. x ph  ->  ph )
3 ax-4 2218 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
42, 3syl5 31 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  ps )
)
51, 4mpg 1558 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) )
6 ax-5o 2219 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
75, 6syl 16 1  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1556  ax-4 2218  ax-5o 2219
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