Theorem bj-nnclav 37162

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 205 and pm2.27 43 chained using syl 18. (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 23	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	a2i 15	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  bj-nnclavi  37163  bj-nnclavc  37164  bj-peircestab  37171