Metamath Proof Explorer


Theorem ibibr

Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005) (Proof shortened by Wolf Lammen, 21-Dec-2013)

Ref Expression
Assertion ibibr ( ( 𝜑𝜓 ) ↔ ( 𝜑 → ( 𝜓𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 pm5.501 ( 𝜑 → ( 𝜓 ↔ ( 𝜑𝜓 ) ) )
2 bicom ( ( 𝜑𝜓 ) ↔ ( 𝜓𝜑 ) )
3 1 2 bitrdi ( 𝜑 → ( 𝜓 ↔ ( 𝜓𝜑 ) ) )
4 3 pm5.74i ( ( 𝜑𝜓 ) ↔ ( 𝜑 → ( 𝜓𝜑 ) ) )