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Theorem ixpeq1d 6871
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 6870 . 2  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
31, 2syl 15 1  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633   X_cixp 6860
This theorem is referenced by:  prdsval  13404  isfunc  13787  funcpropd  13823  natfval  13869  natpropd  13899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-fn 5295  df-ixp 6861
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