Theorem xrneq12i

Description: Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020)

	Ref	Expression
Hypotheses	xrneq12i.1	⊢ 𝐴 = 𝐵
	xrneq12i.2	⊢ 𝐶 = 𝐷
Assertion	xrneq12i	⊢ ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		xrneq12i.1	⊢ 𝐴 = 𝐵
2		xrneq12i.2	⊢ 𝐶 = 𝐷
3		xrneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 )