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Metamath Proof Explorer

Theorem eqeqan12rd

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994)

Ref Expression
Hypotheses eqeqan12rd.1 ⊢ φ → A = B
eqeqan12rd.2 ⊢ ψ → C = D
Assertion eqeqan12rd ⊢ ψ ∧ φ → A = C ↔ B = D

Proof

Step Hyp Ref Expression
1 eqeqan12rd.1 ⊢ φ → A = B
2 eqeqan12rd.2 ⊢ ψ → C = D
3 1 2 eqeqan12d ⊢ φ ∧ ψ → A = C ↔ B = D
4 3 ancoms ⊢ ψ ∧ φ → A = C ↔ B = D