Theorem xrneq12d

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021)

	Ref	Expression
Hypotheses	xrneq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	xrneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	xrneq12d	⊢ ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		xrneq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		xrneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		xrneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )