Theorem bj-nnclav 34653

Description: When ⊥ is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent, which is therefore a minimalistic tautology. Notice the non-intuitionistic proof from peirce 201 and pm2.27 42 chained using syl 17. (Contributed by BJ, 4-Dec-2023.)

Assertion

Ref	Expression
bj-nnclav	⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))

Proof of Theorem bj-nnclav

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	a2i 14	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))

Syntax hints:   → wi 4

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

This theorem is referenced by:  bj-nnclavi  34654  bj-nnclavc  34655  bj-peircestab  34660