Theorem frege54cor0a 37177

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 37144	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	anim2i 591	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) → ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
3		con4 111	. . . 4 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓))
4	3	anim2i 591	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)) → ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
5	2, 4	impbii 198	. 2 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
6		dfbi2 658	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
7		dfifp2 1008	. 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
8	5, 6, 7	3bitr4i 291	1 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 37144

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  frege54cor1a  37178  frege55lem1a  37180  frege55lem2a  37181