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Theorem ixpeq2dv 7070
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dv.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
ixpeq2dv  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem ixpeq2dv
StepHypRef Expression
1 ixpeq2dv.1 . . 3  |-  ( ph  ->  B  =  C )
21adantr 452 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32ixpeq2dva 7069 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   X_cixp 7055
This theorem is referenced by:  prdsval  13670  brssc  14006  isfunc  14053  natfval  14135  isnat  14136  dprdval  15553  elpt  17596  elptr  17597  dfac14  17642
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-in 3319  df-ss 3326  df-ixp 7056
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