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Theorem ixpeq2dv 7404
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 463 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32ixpeq2dva 7403 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   X_cixp 7388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-in 3396  df-ss 3403  df-ixp 7389
This theorem is referenced by:  prdsval  14862  brssc  15220  isfunc  15270  natfval  15352  isnat  15353  dprdval  17147  dprdvalOLD  17149  elpt  20158  elptr  20159  dfac14  20204
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