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Theorem ixpeq2 8465
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
ixpeq2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)

Proof of Theorem ixpeq2
StepHypRef Expression
1 ss2ixp 8464 . . 3 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
2 ss2ixp 8464 . . 3 (∀𝑥𝐴 𝐶𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐵)
31, 2anim12i 615 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
4 eqss 3967 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 3160 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 3165 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 278 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 3967 . 2 (X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶 ↔ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
93, 7, 83imtr4i 295 1 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wral 3133  wss 3919  Xcixp 8451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-in 3926  df-ss 3936  df-ixp 8452
This theorem is referenced by:  ixpeq2dva  8466  ixpint  8479  prdsbas3  16750  pwsbas  16756  ptbasfi  22182  ptunimpt  22196  pttopon  22197  ptcld  22214  ptrescn  22240  ptuncnv  22408  ptunhmeo  22409  ptrest  34966  prdstotbnd  35142  ixpeq2d  41559  hoidmv1le  43096
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