ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpeq1d GIF version

Theorem xpeq1d 4565
Description: Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
Hypothesis
Ref Expression
xpeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
xpeq1d (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Proof of Theorem xpeq1d
StepHypRef Expression
1 xpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 xpeq1 4556 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
31, 2syl 14 1 (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   × cxp 4540
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-opab 3993  df-xp 4548
This theorem is referenced by:  xpssres  4857  ixpsnf1o  6633  xpfi  6821  hashxp  10596
  Copyright terms: Public domain W3C validator