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Metamath Proof Explorer

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