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Theorem ixpeq2dva 8952
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 3146 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 ixpeq2 8951 . 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 395   = wceq 1540  wcel 2108  wral 3061  Xcixp 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-ss 3968  df-ixp 8938
This theorem is referenced by:  ixpeq2dv  8953  dfac9  10177  xpsrnbas  17616  funcpropd  17947  natpropd  18024  prdsmgp  20148  frlmip  21798  elptr2  23582  dfac14  23626  xkoptsub  23662  prdsxmslem2  24542  rrxip  25424  ptrest  37626  prdsbnd2  37802  hoidmvlelem3  46612  ovnhoilem1  46616  ovnhoilem2  46617  hoicoto2  46620  ovnlecvr2  46625  ovncvr2  46626  ovnovollem1  46671  ovnovollem2  46672  hoimbl2  46680  vonhoire  46687  iccvonmbllem  46693  vonioolem2  46696  vonicclem2  46699  vonn0ioo2  46705  vonn0icc2  46707
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