Theorem frege54cor0a 41471

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 41438	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	anim2i 617	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) → ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
3		con4 113	. . . 4 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓))
4	3	anim2i 617	. . 3 ⊢ (((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)) → ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
5	2, 4	impbii 208	. 2 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
6		dfbi2 475	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)))
7		dfifp2 1062	. 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
8	5, 6, 7	3bitr4i 303	1 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  if-wif 1060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 41438

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061

This theorem is referenced by:  frege54cor1a  41472  frege55lem1a  41474  frege55lem2a  41475