Theorem curryax 890

Description: A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 891 (obtained by substituting ⊥ for 𝜓) but positive. For another non-intuitionistic positive statement, see peirce 204. (Contributed by BJ, 4-Apr-2021.)

Assertion

Ref	Expression
curryax	⊢ (𝜑 ∨ (𝜑 → 𝜓))

Proof of Theorem curryax

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2	1	orri 858	1 ⊢ (𝜑 ∨ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∨ 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:  reuprg  4639  tsim2  35424