Definition df-bi-D2.1 107

Description: Definition of Biconditional, '↔'. Read as "if and only if".

Assertion

Ref	Expression
df-bi-D2.1	⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))

Detailed syntax breakdown of Definition df-bi-D2.1

Step	Hyp	Ref	Expression
1		wff-ph	. . . . 5 wff 𝜑
2		wff-ps	. . . . 5 wff 𝜓
3	1, 2	wff-bi 104	. . . 4 wff (𝜑 ↔ 𝜓)
4	1, 2	wff-imp 10	. . . . . 6 wff (𝜑 → 𝜓)
5	2, 1	wff-imp 10	. . . . . . 7 wff (𝜓 → 𝜑)
6	5	wff-neg 9	. . . . . 6 wff ¬ (𝜓 → 𝜑)
7	4, 6	wff-imp 10	. . . . 5 wff ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))
8	7	wff-neg 9	. . . 4 wff ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))
9	3, 8	wff-imp 10	. . 3 wff ((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
10	8, 3	wff-imp 10	. . . 4 wff (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
11	10	wff-neg 9	. . 3 wff ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
12	9, 11	wff-imp 10	. 2 wff (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
13	12	wff-neg 9	1 wff ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))

This definition is referenced by:  dfbiif-P2.3a  108  dfbionlyif-P2.3b  109  dfbialt-P2.4  110