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Theorem xpeq1 4359
Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
xpeq1 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Proof of Theorem xpeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2101 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 438 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
32opabbidv 3823 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
4 df-xp 4351 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
5 df-xp 4351 . 2 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
63, 4, 53eqtr4g 2097 1 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  {copab 3817   × cxp 4343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-opab 3819  df-xp 4351
This theorem is referenced by:  xpeq12  4364  xpeq1i  4365  xpeq1d  4368  opthprc  4391  reseq2  4607  xpeq0r  4746  xpdisj1  4747  xpima1  4767  xpsneng  6296  xpcomeng  6302  xpdom2g  6306
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