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Theorem ixpeq2dv 6661
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dv.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dv (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dv
StepHypRef Expression
1 ixpeq2dv.1 . . 3 (𝜑𝐵 = 𝐶)
21adantr 274 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ixpeq2dva 6660 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Xcixp 6645
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-in 3108  df-ss 3115  df-ixp 6646
This theorem is referenced by: (None)
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