Theorem xpeq12

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

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

Proof

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