Theorem xrneq2d

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

	Ref	Expression
Hypothesis	xrneq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	xrneq2d	⊢ ( 𝜑 → ( 𝐶 ⋉ 𝐴 ) = ( 𝐶 ⋉ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xrneq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		xrneq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⋉ 𝐴 ) = ( 𝐶 ⋉ 𝐵 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐶 ⋉ 𝐴 ) = ( 𝐶 ⋉ 𝐵 ) )