Metamath Proof Explorer

Theorem pm5.32

Description: Distribution of implication over biconditional. Theorem *5.32 of WhiteheadRussell p. 125. (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion pm5.32 ⊢ φ → ψ ↔ χ ↔ φ ∧ ψ ↔ φ ∧ χ

Proof

Step Hyp Ref Expression
1 notbi ⊢ ψ ↔ χ ↔ ¬ ψ ↔ ¬ χ
2 1 imbi2i ⊢ φ → ψ ↔ χ ↔ φ → ¬ ψ ↔ ¬ χ
3 pm5.74 ⊢ φ → ¬ ψ ↔ ¬ χ ↔ φ → ¬ ψ ↔ φ → ¬ χ
4 notbi ⊢ φ → ¬ ψ ↔ φ → ¬ χ ↔ ¬ φ → ¬ ψ ↔ ¬ φ → ¬ χ
5 2 3 4 3bitri ⊢ φ → ψ ↔ χ ↔ ¬ φ → ¬ ψ ↔ ¬ φ → ¬ χ
6 df-an ⊢ φ ∧ ψ ↔ ¬ φ → ¬ ψ
7 df-an ⊢ φ ∧ χ ↔ ¬ φ → ¬ χ
8 6 7 bibi12i ⊢ φ ∧ ψ ↔ φ ∧ χ ↔ ¬ φ → ¬ ψ ↔ ¬ φ → ¬ χ
9 5 8 bitr4i ⊢ φ → ψ ↔ χ ↔ φ ∧ ψ ↔ φ ∧ χ