Theorem simprimi 44961

Description: Inference associated with simprim 166. Proved exactly as step 11 is obtained from step 4 in dfbi1ALTa 44960. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
simprimi.1	⊢ ¬ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
simprimi	⊢ 𝜓

Proof of Theorem simprimi

Step	Hyp	Ref	Expression
1		tru 1544	. 2 ⊢ ⊤
2		ax-1 6	. . . 4 ⊢ (¬ 𝜓 → (𝜑 → ¬ 𝜓))
3		simprimi.1	. . . . . . . 8 ⊢ ¬ (𝜑 → ¬ 𝜓)
4	3	a1i 11	. . . . . . 7 ⊢ (¬ ¬ ⊤ → ¬ (𝜑 → ¬ 𝜓))
5	4	con4i 114	. . . . . 6 ⊢ ((𝜑 → ¬ 𝜓) → ¬ ⊤)
6	5	a1i 11	. . . . 5 ⊢ (¬ 𝜓 → ((𝜑 → ¬ 𝜓) → ¬ ⊤))
7	6	a2i 14	. . . 4 ⊢ ((¬ 𝜓 → (𝜑 → ¬ 𝜓)) → (¬ 𝜓 → ¬ ⊤))
8	2, 7	ax-mp 5	. . 3 ⊢ (¬ 𝜓 → ¬ ⊤)
9	8	con4i 114	. 2 ⊢ (⊤ → 𝜓)
10	1, 9	ax-mp 5	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4  ⊤wtru 1541

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 207  df-tru 1543

This theorem is referenced by:  dfbi1ALTb  44962