Theorem poe-P1.11b.AC.2SH 67

Description: Deductive Form of poe-P1.11b 66.

Hypotheses

Ref	Expression
poe-P1.11b.AC.2SH.1	⊢ (𝛾 → 𝜑)
poe-P1.11b.AC.2SH.2	⊢ (𝛾 → ¬ 𝜑)

Assertion

Ref	Expression
poe-P1.11b.AC.2SH	⊢ (𝛾 → 𝜓)

Proof of Theorem poe-P1.11b.AC.2SH

Step	Hyp	Ref	Expression
1		poe-P1.11b.AC.2SH.2	. 2 ⊢ (𝛾 → ¬ 𝜑)
2		poe-P1.11b.AC.2SH.1	. . . 4 ⊢ (𝛾 → 𝜑)
3		poe-P1.11b 66	. . . . . 6 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
4	3	axL1.SH 30	. . . . 5 ⊢ (𝛾 → (𝜑 → (¬ 𝜑 → 𝜓)))
5	4	rcp-FR1.SH 40	. . . 4 ⊢ ((𝛾 → 𝜑) → (𝛾 → (¬ 𝜑 → 𝜓)))
6	2, 5	ax-MP 14	. . 3 ⊢ (𝛾 → (¬ 𝜑 → 𝜓))
7	6	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → ¬ 𝜑) → (𝛾 → 𝜓))
8	1, 7	ax-MP 14	1 ⊢ (𝛾 → 𝜓)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem is referenced by:  ndnege-P3.4  169