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Theorem xpeq1 4302
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 2098 . . . 4 (A = B → (x Ax B))
21anbi1d 438 . . 3 (A = B → ((x A y 𝐶) ↔ (x B y 𝐶)))
32opabbidv 3814 . 2 (A = B → {⟨x, y⟩ ∣ (x A y 𝐶)} = {⟨x, y⟩ ∣ (x B y 𝐶)})
4 df-xp 4294 . 2 (A × 𝐶) = {⟨x, y⟩ ∣ (x A y 𝐶)}
5 df-xp 4294 . 2 (B × 𝐶) = {⟨x, y⟩ ∣ (x B y 𝐶)}
63, 4, 53eqtr4g 2094 1 (A = B → (A × 𝐶) = (B × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {copab 3808   × cxp 4286
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpeq12  4307  xpeq1i  4308  xpeq1d  4311  opthprc  4334  reseq2  4550  xpeq0r  4689  xpdisj1  4690  xpima1  4710  xpsneng  6232  xpcomeng  6238  xpdom2g  6242
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