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Metamath Proof Explorer

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 )