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Theorem ixpeq2dva 6613
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dva.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
ixpeq2dva  |-  ( 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 ixpeq2dva
StepHypRef Expression
1 ixpeq2dva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
21ralrimiva 2508 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 ixpeq2 6612 . 2  |-  ( A. x  e.  A  B  =  C  ->  X_ x  e.  A  B  =  X_ x  e.  A  C
)
42, 3syl 14 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   A.wral 2417   X_cixp 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-in 3080  df-ss 3087  df-ixp 6599
This theorem is referenced by:  ixpeq2dv  6614
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