Metamath Proof Explorer


Theorem pm5.74rd

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997)

Ref Expression
Hypothesis pm5.74rd.1 ( 𝜑 → ( ( 𝜓𝜒 ) ↔ ( 𝜓𝜃 ) ) )
Assertion pm5.74rd ( 𝜑 → ( 𝜓 → ( 𝜒𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 pm5.74rd.1 ( 𝜑 → ( ( 𝜓𝜒 ) ↔ ( 𝜓𝜃 ) ) )
2 pm5.74 ( ( 𝜓 → ( 𝜒𝜃 ) ) ↔ ( ( 𝜓𝜒 ) ↔ ( 𝜓𝜃 ) ) )
3 1 2 sylibr ( 𝜑 → ( 𝜓 → ( 𝜒𝜃 ) ) )