Theorem frege54cor0a

Description: Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege54cor0a	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege28	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )
2	1	anim2i	⊢ ( ( ( 𝜓 → 𝜑 ) ∧ ( 𝜑 → 𝜓 ) ) → ( ( 𝜓 → 𝜑 ) ∧ ( ¬ 𝜓 → ¬ 𝜑 ) ) )
3		con4	⊢ ( ( ¬ 𝜓 → ¬ 𝜑 ) → ( 𝜑 → 𝜓 ) )
4	3	anim2i	⊢ ( ( ( 𝜓 → 𝜑 ) ∧ ( ¬ 𝜓 → ¬ 𝜑 ) ) → ( ( 𝜓 → 𝜑 ) ∧ ( 𝜑 → 𝜓 ) ) )
5	2 4	impbii	⊢ ( ( ( 𝜓 → 𝜑 ) ∧ ( 𝜑 → 𝜓 ) ) ↔ ( ( 𝜓 → 𝜑 ) ∧ ( ¬ 𝜓 → ¬ 𝜑 ) ) )
6		dfbi2	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ ( ( 𝜓 → 𝜑 ) ∧ ( 𝜑 → 𝜓 ) ) )
7		dfifp2	⊢ ( if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) ↔ ( ( 𝜓 → 𝜑 ) ∧ ( ¬ 𝜓 → ¬ 𝜑 ) ) )
8	5 6 7	3bitr4i	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ if- ( 𝜓 , 𝜑 , ¬ 𝜑 ) )