Theorem xpeq12d 4476

Description: Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)

Hypotheses

Ref	Expression
xpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
xpeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
xpeq12d	⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Proof of Theorem xpeq12d

Step	Hyp	Ref	Expression
1		xpeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		xpeq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		xpeq12 4470	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	syl2anc 404	1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   = wceq 1290   × cxp 4449

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-opab 3906  df-xp 4457

This theorem is referenced by:  opeliunxp  4506  mpt2mptsx  5981  dmmpt2ssx  5983  fmpt2x  5984  disjxp1  6015  erssxp  6329  fsum2dlemstep  10882  fisumcom2  10886