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Theorem fveleq 24962
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
Assertion
Ref Expression
fveleq  |-  ( A  =  B  ->  (
( ph  ->  ( F `
 A )  e.  P )  <->  ( ph  ->  ( F `  B
)  e.  P ) ) )

Proof of Theorem fveleq
StepHypRef Expression
1 fveq2 5541 . . 3  |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
21eleq1d 2362 . 2  |-  ( A  =  B  ->  (
( F `  A
)  e.  P  <->  ( F `  B )  e.  P
) )
32imbi2d 307 1  |-  ( A  =  B  ->  (
( ph  ->  ( F `
 A )  e.  P )  <->  ( ph  ->  ( F `  B
)  e.  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   ` cfv 5271
This theorem is referenced by:  findfvcl  24963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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