Theorem xpeq12d

Description: Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013)

	Ref	Expression
Hypotheses	xpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	xpeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	xpeq12d	⊢ ( 𝜑 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		xpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		xpeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		xpeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐷 ) )