Metamath Proof Explorer


Theorem ifpbi13

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020)

Ref Expression
Assertion ifpbi13 ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ if- ( 𝜓 , 𝜏 , 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 simpl ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( 𝜑𝜓 ) )
2 1 imbi1d ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( ( 𝜑𝜏 ) ↔ ( 𝜓𝜏 ) ) )
3 notbi ( ( 𝜑𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4 imbi12 ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ( 𝜒𝜃 ) → ( ( ¬ 𝜑𝜒 ) ↔ ( ¬ 𝜓𝜃 ) ) ) )
5 3 4 sylbi ( ( 𝜑𝜓 ) → ( ( 𝜒𝜃 ) → ( ( ¬ 𝜑𝜒 ) ↔ ( ¬ 𝜓𝜃 ) ) ) )
6 5 imp ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( ( ¬ 𝜑𝜒 ) ↔ ( ¬ 𝜓𝜃 ) ) )
7 2 6 anbi12d ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( ( ( 𝜑𝜏 ) ∧ ( ¬ 𝜑𝜒 ) ) ↔ ( ( 𝜓𝜏 ) ∧ ( ¬ 𝜓𝜃 ) ) ) )
8 dfifp2 ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ ( ( 𝜑𝜏 ) ∧ ( ¬ 𝜑𝜒 ) ) )
9 dfifp2 ( if- ( 𝜓 , 𝜏 , 𝜃 ) ↔ ( ( 𝜓𝜏 ) ∧ ( ¬ 𝜓𝜃 ) ) )
10 7 8 9 3bitr4g ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) → ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ if- ( 𝜓 , 𝜏 , 𝜃 ) ) )