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Theorem ixpeq1d 7074
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ixpeq1d  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem ixpeq1d
StepHypRef Expression
1 ixpeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ixpeq1 7073 . 2  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
31, 2syl 16 1  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   X_cixp 7063
This theorem is referenced by:  elixpsn  7101  ixpsnf1o  7102  dfac9  8016  prdsval  13678  isfunc  14061  funcpropd  14097  natfval  14143  natpropd  14173  dprdval  15561  ptval  17602  dfac14  17650  ptuncnv  17839  ptunhmeo  17840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-fn 5457  df-ixp 7064
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