MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixpeq1d Structured version   Visualization version   GIF version

Theorem ixpeq1d 8903
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ixpeq1d (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq1d
StepHypRef Expression
1 ixpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ixpeq1 8902 . 2 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
31, 2syl 17 1 (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  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-rex 3072  df-fn 6547  df-ixp 8892
This theorem is referenced by:  elixpsn  8931  ixpsnf1o  8932  dfac9  10131  prdsval  17401  isfunc  17814  funcpropd  17851  natfval  17897  natpropd  17929  dprdval  19873  ptval  23074  dfac14  23122  ptuncnv  23311  ptunhmeo  23312  hoidmvle  45316  hoimbl  45347
  Copyright terms: Public domain W3C validator