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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