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Theorem ixpeq2dv 8078
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dv.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dv (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dv
StepHypRef Expression
1 ixpeq2dv.1 . . 3 (𝜑𝐵 = 𝐶)
21adantr 472 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ixpeq2dva 8077 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  Xcixp 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-in 3710  df-ss 3717  df-ixp 8063
This theorem is referenced by:  prdsval  16288  brssc  16646  isfunc  16696  natfval  16778  isnat  16779  dprdval  18573  elpt  21548  elptr  21549  dfac14  21594  hoicvrrex  41245  ovncvrrp  41253  ovnsubaddlem1  41259  ovnsubadd  41261  hoidmvlelem3  41286  hoidmvle  41289  ovnhoilem1  41290  ovnhoilem2  41291  ovnhoi  41292  hspval  41298  ovncvr2  41300  hspmbllem2  41316  hspmbl  41318  hoimbl  41320  opnvonmbl  41323  ovnovollem1  41345  ovnovollem3  41347  iinhoiicclem  41362  iinhoiicc  41363  vonioolem2  41370  vonioo  41371  vonicclem2  41373  vonicc  41374
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