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Theorem fnopabeqd 30372
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 2471 . . 3  |-  ( ph  ->  ( y  =  B  <-> 
y  =  C ) )
32anbi2d 703 . 2  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  C ) ) )
43opabbidv 4520 1  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {copab 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-opab 4516
This theorem is referenced by: (None)
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