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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 ) ) )