Theorem xpeq1d

Description: Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010)

	Ref	Expression
Hypothesis	xpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	xpeq1d	⊢ ( 𝜑 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		xpeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 × 𝐶 ) = ( 𝐵 × 𝐶 ) )