Metamath Proof Explorer


Theorem wl-ifp4impr

Description: If one case of an if- condition is a consequence of the other, the expression in dfifp4 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024)

Ref Expression
Assertion wl-ifp4impr ( ( 𝜒𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑𝜒 ) ∧ 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 wl-ifpimpr ( ( 𝜒𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑𝜓 ) ∨ 𝜒 ) ) )
2 pm4.71 ( ( 𝜒𝜓 ) ↔ ( 𝜒 ↔ ( 𝜒𝜓 ) ) )
3 2 biimpi ( ( 𝜒𝜓 ) → ( 𝜒 ↔ ( 𝜒𝜓 ) ) )
4 3 orbi2d ( ( 𝜒𝜓 ) → ( ( ( 𝜑𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜑𝜓 ) ∨ ( 𝜒𝜓 ) ) ) )
5 andir ( ( ( 𝜑𝜒 ) ∧ 𝜓 ) ↔ ( ( 𝜑𝜓 ) ∨ ( 𝜒𝜓 ) ) )
6 4 5 bitr4di ( ( 𝜒𝜓 ) → ( ( ( 𝜑𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜑𝜒 ) ∧ 𝜓 ) ) )
7 1 6 bitrd ( ( 𝜒𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑𝜒 ) ∧ 𝜓 ) ) )