Theorem frege54cor0a 44863

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

Assertion

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

Proof of Theorem frege54cor0a

Step	Hyp	Ref	Expression
1		ax-frege28 44830	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	anim2i 629	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) → ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
3		con4 114	. . . 4 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓))
4	3	anim2i 629	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)) → ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
5	2, 4	impbii 212	. 2 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
6		dfbi2 480	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
7		dfifp2 1080	. 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
8	5, 6, 7	3bitr4i 306	1 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 44830

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  frege54cor1a  44864  frege55lem1a  44866  frege55lem2a  44867