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Theorem ixpeq2d 42195
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 416 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3129 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 ixpeq2 8534 . 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 399   = wceq 1542  wnf 1790  wcel 2114  wral 3054  Xcixp 8520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-v 3402  df-in 3860  df-ss 3870  df-ixp 8521
This theorem is referenced by:  hoicvrrex  43677  ovnlecvr  43679  ovnhoilem1  43722  hoi2toco  43728  ovnlecvr2  43731  opnvonmbllem1  43753
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