Metamath Proof Explorer

Theorem frege53a

Description: Lemma for frege55a . Proposition 53 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege53a ⊢ if- φ θ χ → φ ↔ ψ → if- ψ θ χ

Proof

Step Hyp Ref Expression
1 ax-frege52a ⊢ φ ↔ ψ → if- φ θ χ → if- ψ θ χ
2 ax-frege8 ⊢ φ ↔ ψ → if- φ θ χ → if- ψ θ χ → if- φ θ χ → φ ↔ ψ → if- ψ θ χ
3 1 2 ax-mp ⊢ if- φ θ χ → φ ↔ ψ → if- ψ θ χ