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Metamath Proof Explorer

Theorem ineq12d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Hypotheses ineq1d.1 ⊢ φ → A = B
ineq12d.2 ⊢ φ → C = D
Assertion ineq12d ⊢ φ → A ∩ C = B ∩ D

Proof

Step Hyp Ref Expression
1 ineq1d.1 ⊢ φ → A = B
2 ineq12d.2 ⊢ φ → C = D
3 ineq12 ⊢ A = B ∧ C = D → A ∩ C = B ∩ D
4 1 2 3 syl2anc ⊢ φ → A ∩ C = B ∩ D