Theorem confun3 43171

Description: Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020.)

Hypotheses

Ref	Expression
confun3.1	⊢ (𝜑 ↔ (𝜒 → 𝜓))
confun3.2	⊢ (𝜃 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))
confun3.3	⊢ (𝜒 → 𝜓)
confun3.4	⊢ (𝜒 → ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))
confun3.5	⊢ ((𝜒 → 𝜓) → ((𝜒 → 𝜓) → 𝜓))

Assertion

Ref	Expression
confun3	⊢ (𝜒 → (¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)) ↔ (𝜒 → 𝜓)))

Proof of Theorem confun3

Step	Hyp	Ref	Expression
1		confun3.3	. 2 ⊢ (𝜒 → 𝜓)
2		confun3.4	. 2 ⊢ (𝜒 → ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))
3		confun3.5	. 2 ⊢ ((𝜒 → 𝜓) → ((𝜒 → 𝜓) → 𝜓))
4	1, 1, 2, 3	confun 43169	1 ⊢ (𝜒 → (¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)) ↔ (𝜒 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398

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

This theorem depends on definitions:  df-bi 209

This theorem is referenced by: (None)