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Theorem ax5 2222
Description: Rederivation of axiom ax-5 1566 from ax-5o 2212 and other older axioms. See ax5o 1765 for the derivation of ax-5o 2212 from ax-5 1566. (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 2212 . . 3  |-  ( A. x ( A. x
( ph  ->  ps )  ->  ( A. x ph  ->  ps ) )  -> 
( A. x (
ph  ->  ps )  ->  A. x ( A. x ph  ->  ps ) ) )
2 ax-4 2211 . . . 4  |-  ( A. x ph  ->  ph )
3 ax-4 2211 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
42, 3syl5 30 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  ps )
)
51, 4mpg 1557 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) )
6 ax-5o 2212 . 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 1549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 8  ax-gen 1555  ax-4 2211  ax-5o 2212
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