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Theorem fveleq 24266
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 5458 . . 3  |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
21eleq1d 2324 . 2  |-  ( A  =  B  ->  (
( F `  A
)  e.  P  <->  ( F `  B )  e.  P
) )
32imbi2d 309 1  |-  ( A  =  B  ->  (
( ph  ->  ( F `
 A )  e.  P )  <->  ( ph  ->  ( F `  B
)  e.  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   ` cfv 4673
This theorem is referenced by:  findfvcl  24267
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689
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