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Theorem xpeq2 4387
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 2117 . . . 4 (𝐴 = 𝐵 → (𝑦𝐴𝑦𝐵))
21anbi2d 445 . . 3 (𝐴 = 𝐵 → ((𝑥𝐶𝑦𝐴) ↔ (𝑥𝐶𝑦𝐵)))
32opabbidv 3850 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)})
4 df-xp 4378 . 2 (𝐶 × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)}
5 df-xp 4378 . 2 (𝐶 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)}
63, 4, 53eqtr4g 2113 1 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {copab 3844   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-opab 3846  df-xp 4378
This theorem is referenced by:  xpeq12  4391  xpeq2i  4393  xpeq2d  4396  xpeq0r  4773  xpdisj2  4775  xpcomeng  6332
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