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Theorem ixpeq2d 38262
 Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ixpeq2d.1 𝑥𝜑
ixpeq2d.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2d (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)

Proof of Theorem ixpeq2d
StepHypRef Expression
1 ixpeq2d.1 . . 3 𝑥𝜑
2 ixpeq2d.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ex 449 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 ixpeq2 7808 . 2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-in 3547  df-ss 3554  df-ixp 7795 This theorem is referenced by:  hoicvrrex  39446  ovnlecvr  39448  ovnhoilem1  39491  hoi2toco  39497  ovnlecvr2  39500  opnvonmbllem1  39522
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