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Theorem ixpeq2 8950
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 8949 . . 3 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
2 ss2ixp 8949 . . 3 (∀𝑥𝐴 𝐶𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐵)
31, 2anim12i 613 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
4 eqss 4011 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 3091 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 3109 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 275 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 4011 . 2 (X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶 ↔ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
93, 7, 83imtr4i 292 1 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wral 3059  wss 3963  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-ss 3980  df-ixp 8937
This theorem is referenced by:  ixpeq2dva  8951  ixpint  8964  prdsbas3  17528  pwsbas  17534  ptbasfi  23605  ptunimpt  23619  pttopon  23620  ptcld  23637  ptrescn  23663  ptuncnv  23831  ptunhmeo  23832  ixpeq12i  36183  ptrest  37606  prdstotbnd  37781  ixpeq2d  45008  hoidmv1le  46550
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