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Theorem ixpeq2d 40890
 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 413 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3185 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 ixpeq2 8331 . 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 396   = wceq 1525  Ⅎwnf 1769   ∈ wcel 2083  ∀wral 3107  Xcixp 8317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-in 3872  df-ss 3880  df-ixp 8318 This theorem is referenced by:  hoicvrrex  42402  ovnlecvr  42404  ovnhoilem1  42447  hoi2toco  42453  ovnlecvr2  42456  opnvonmbllem1  42478
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