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Theorem fveleq 28434
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 5792 . . 3  |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
21eleq1d 2520 . 2  |-  ( A  =  B  ->  (
( F `  A
)  e.  P  <->  ( F `  B )  e.  P
) )
32imbi2d 316 1  |-  ( A  =  B  ->  (
( ph  ->  ( F `
 A )  e.  P )  <->  ( ph  ->  ( F `  B
)  e.  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527
This theorem is referenced by:  findfvcl  28435
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