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Theorem xpeq1 4521
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 2179 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 458 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
32opabbidv 3962 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
4 df-xp 4513 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
5 df-xp 4513 . 2 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
63, 4, 53eqtr4g 2173 1 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  {copab 3956   × cxp 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-opab 3958  df-xp 4513
This theorem is referenced by:  xpeq12  4526  xpeq1i  4527  xpeq1d  4530  opthprc  4558  reseq2  4782  xpeq0r  4929  xpdisj1  4931  xpima1  4953  pmvalg  6519  xpsneng  6682  xpcomeng  6688  xpdom2g  6692  xpfi  6784  exmidomni  6980  exmidfodomrlemim  7021  hashxp  10512  txuni2  12320  txbas  12322  txopn  12329  txrest  12340  txdis  12341  txdis1cn  12342  xmettxlem  12573  xmettx  12574
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