This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pm5.32d

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996)

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

Proof

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