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Theorem ixpeq2dva 8906
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dva.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dva (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dva
StepHypRef Expression
1 ixpeq2dva.1 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
21ralrimiva 3147 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 ixpeq2 8905 . 2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
42, 3syl 17 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Xcixp 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966  df-ixp 8892
This theorem is referenced by:  ixpeq2dv  8907  dfac9  10131  xpsrnbas  17517  funcpropd  17851  natpropd  17929  prdsmgp  20132  frlmip  21333  elptr2  23078  dfac14  23122  xkoptsub  23158  prdsxmslem2  24038  rrxip  24907  ptrest  36487  prdsbnd2  36663  hoidmvlelem3  45313  ovnhoilem1  45317  ovnhoilem2  45318  hoicoto2  45321  ovnlecvr2  45326  ovncvr2  45327  ovnovollem1  45372  ovnovollem2  45373  hoimbl2  45381  vonhoire  45388  iccvonmbllem  45394  vonioolem2  45397  vonicclem2  45400  vonn0ioo2  45406  vonn0icc2  45408
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