Theorem xpeq12 4804

Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
xpeq12	⊢ ((A = B ∧ C = D) → (A × C) = (B × D))

Proof of Theorem xpeq12

Step	Hyp	Ref	Expression
1		xpeq1 4799	. 2 ⊢ (A = B → (A × C) = (B × C))
2		xpeq2 4800	. 2 ⊢ (C = D → (B × C) = (B × D))
3	1, 2	sylan9eq 2405	1 ⊢ ((A = B ∧ C = D) → (A × C) = (B × D))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-opab 4624  df-xp 4785

This theorem is referenced by:  xpeq12i  4807  xpeq12d  4810  xpid11  4927  xp11  5057  fnpprod  5844  mucnc  6132