Theorem xrneq1d

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021)

	Ref	Expression
Hypothesis	xrneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	xrneq1d	⊢ ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xrneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		xrneq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )