Theorem xpeq12d 4705

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 4699	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   = wceq 1373   × cxp 4678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-opab 4111  df-xp 4686

This theorem is referenced by:  sqxpeqd  4706  opeliunxp  4735  mpomptsx  6293  dmmpossx  6295  fmpox  6296  disjxp1  6332  erssxp  6653  cc2lem  7391  cc2  7392  fsum2dlemstep  11795  fisumcom2  11799  fprod2dlemstep  11983  fprodcom2fi  11987  psrval  14478  txbas  14780