Theorem xpeq12d 4613

Description: Equality deduction for Cartesian 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 4607	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	syl2anc 409	1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   = wceq 1335   × cxp 4586

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-opab 4028  df-xp 4594

This theorem is referenced by:  sqxpeqd  4614  opeliunxp  4643  mpomptsx  6147  dmmpossx  6149  fmpox  6150  disjxp1  6185  erssxp  6505  cc2lem  7188  cc2  7189  fsum2dlemstep  11342  fisumcom2  11346  fprod2dlemstep  11530  fprodcom2fi  11534  txbas  12728