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Theorem ixpeq2 7882
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 7881 . . 3 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
2 ss2ixp 7881 . . 3 (∀𝑥𝐴 𝐶𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐵)
31, 2anim12i 589 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
4 eqss 3603 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 2976 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 3059 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 264 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 3603 . 2 (X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶 ↔ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
93, 7, 83imtr4i 281 1 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wral 2908  wss 3560  Xcixp 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-in 3567  df-ss 3574  df-ixp 7869
This theorem is referenced by:  ixpeq2dva  7883  ixpint  7895  prdsbas3  16081  pwsbas  16087  ptbasfi  21324  ptunimpt  21338  pttopon  21339  ptcld  21356  ptrescn  21382  ptuncnv  21550  ptunhmeo  21551  ptrest  33079  prdstotbnd  33264  ixpeq2d  38759  hoidmv1le  40145
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