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Theorem ixpeq2 8770
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 8769 . . 3 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
2 ss2ixp 8769 . . 3 (∀𝑥𝐴 𝐶𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐵)
31, 2anim12i 613 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
4 eqss 3947 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 3092 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 3110 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 274 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 3947 . 2 (X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶 ↔ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶X𝑥𝐴 𝐶X𝑥𝐴 𝐵))
93, 7, 83imtr4i 291 1 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wral 3061  wss 3898  Xcixp 8756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3443  df-in 3905  df-ss 3915  df-ixp 8757
This theorem is referenced by:  ixpeq2dva  8771  ixpint  8784  prdsbas3  17289  pwsbas  17295  ptbasfi  22838  ptunimpt  22852  pttopon  22853  ptcld  22870  ptrescn  22896  ptuncnv  23064  ptunhmeo  23065  ptrest  35881  prdstotbnd  36057  ixpeq2d  42936  hoidmv1le  44469
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