Theorem orasim-P3.48-L1 359

Description: Lemma for orasim-P3.48a 361 and orasimint-P3.48b 362. †

Assertion

Ref	Expression
orasim-P3.48-L1	⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Proof of Theorem orasim-P3.48-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑 ∧ 𝜑) → 𝜑)
2		rcp-NDASM2of3 196	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑 ∧ 𝜑) → ¬ 𝜑)
3	1, 2	ndnege-P3.4 169	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑 ∧ 𝜑) → 𝜓)
4		rcp-NDASM3of3 197	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑 ∧ 𝜓) → 𝜓)
5		rcp-NDASM1of2 193	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑) → (𝜑 ∨ 𝜓))
6	3, 4, 5	rcp-NDORE3 236	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜑) → 𝜓)
7	6	rcp-NDIMI2 224	1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∨ wff-or 144   ∧ wff-rcp-AND3 160

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

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  orasim-P3.48a  361  orasimint-P3.48b  362