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Theorem fveleq 30144
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 5848 . . 3  |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
21eleq1d 2523 . 2  |-  ( A  =  B  ->  (
( F `  A
)  e.  P  <->  ( F `  B )  e.  P
) )
32imbi2d 314 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 1398    e. wcel 1823   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578
This theorem is referenced by:  findfvcl  30145
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