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Theorem xpeq2 4303
 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 2098 . . . 4 (A = B → (y Ay B))
21anbi2d 437 . . 3 (A = B → ((x 𝐶 y A) ↔ (x 𝐶 y B)))
32opabbidv 3814 . 2 (A = B → {⟨x, y⟩ ∣ (x 𝐶 y A)} = {⟨x, y⟩ ∣ (x 𝐶 y B)})
4 df-xp 4294 . 2 (𝐶 × A) = {⟨x, y⟩ ∣ (x 𝐶 y A)}
5 df-xp 4294 . 2 (𝐶 × B) = {⟨x, y⟩ ∣ (x 𝐶 y B)}
63, 4, 53eqtr4g 2094 1 (A = B → (𝐶 × A) = (𝐶 × B))
 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  xpeq2i  4309  xpeq2d  4312  xpeq0r  4689  xpdisj2  4691  xpcomeng  6238
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