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Theorem xpeq2 4443
Description: Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
Assertion
Ref Expression
xpeq2 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Proof of Theorem xpeq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2151 . . . 4 (𝐴 = 𝐵 → (𝑦𝐴𝑦𝐵))
21anbi2d 452 . . 3 (𝐴 = 𝐵 → ((𝑥𝐶𝑦𝐴) ↔ (𝑥𝐶𝑦𝐵)))
32opabbidv 3896 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)})
4 df-xp 4434 . 2 (𝐶 × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)}
5 df-xp 4434 . 2 (𝐶 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)}
63, 4, 53eqtr4g 2145 1 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {copab 3890   × cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-opab 3892  df-xp 4434
This theorem is referenced by:  xpeq12  4447  xpeq2i  4449  xpeq2d  4452  xpeq0r  4841  xpdisj2  4843  pmvalg  6396  xpcomeng  6524  djueq12  6711
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