Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ixpeq1d GIF version

Theorem ixpeq1d 6611
 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 6610 . 2 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
31, 2syl 14 1 (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  Xcixp 6599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-fn 5133  df-ixp 6600 This theorem is referenced by:  elixpsn  6636  ixpsnf1o  6637
 Copyright terms: Public domain W3C validator