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Metamath Proof Explorer

Theorem pm5.32da

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006)

Ref Expression
Hypothesis pm5.32da.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )
Assertion pm5.32da ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 pm5.32da.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )
2 1 ex ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
3 2 pm5.32d ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )