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Theorem fnopabeqd 26385
Description: Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
Hypothesis
Ref Expression
fnopabeqd.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
fnopabeqd  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem fnopabeqd
StepHypRef Expression
1 fnopabeqd.1 . . . 4  |-  ( ph  ->  B  =  C )
21eqeq2d 2294 . . 3  |-  ( ph  ->  ( y  =  B  <-> 
y  =  C ) )
32anbi2d 684 . 2  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  C ) ) )
43opabbidv 4082 1  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {copab 4076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-opab 4078
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