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Theorem ixpeq2dva 6658
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dva.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dva (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dva
StepHypRef Expression
1 ixpeq2dva.1 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
21ralrimiva 2530 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 ixpeq2 6657 . 2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
42, 3syl 14 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wral 2435  Xcixp 6643
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 6644
This theorem is referenced by:  ixpeq2dv  6659
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