Theorem xpeq12i

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

	Ref	Expression
Hypotheses	xpeq12i.1	⊢ 𝐴 = 𝐵
	xpeq12i.2	⊢ 𝐶 = 𝐷
Assertion	xpeq12i	⊢ ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		xpeq12i.1	⊢ 𝐴 = 𝐵
2		xpeq12i.2	⊢ 𝐶 = 𝐷
3		xpeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 )