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Theorem ixpeq2dv 6538
 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 272 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ixpeq2dva 6537 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  Xcixp 6522 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-in 3027  df-ss 3034  df-ixp 6523 This theorem is referenced by: (None)
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