Theorem rb-imdf 1747

Description: The definition of implication, in terms of ∨ and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rb-imdf	⊢ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓)))

Proof of Theorem rb-imdf

Step	Hyp	Ref	Expression
1		imor 849	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
2		rb-bijust 1746	. 2 ⊢ (((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) ↔ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓))))
3	1, 2	mpbi 232	1 ⊢ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843

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  df-or 844

This theorem is referenced by:  re1axmp  1761  re2luk1  1762  re2luk2  1763  re2luk3  1764