Theorem andasim-P3.46-L2 355

Description: Lemma for andasim-P3.46a 356.

Assertion

Ref	Expression
andasim-P3.46-L2	⊢ (¬ (𝜑 → ¬ 𝜓) → (𝜑 ∧ 𝜓))

Proof of Theorem andasim-P3.46-L2

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . . 6 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜑) → ¬ 𝜑)
2	1	axL1-P3.21 252	. . . . 5 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
3	2	trnsp-P3.31a 279	. . . 4 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜑) → (𝜑 → ¬ 𝜓))
4		rcp-NDASM1of2 193	. . . 4 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜑) → ¬ (𝜑 → ¬ 𝜓))
5	3, 4	rcp-NDNEGI2 219	. . 3 ⊢ (¬ (𝜑 → ¬ 𝜓) → ¬ ¬ 𝜑)
6	5	dnege-P3.30 276	. 2 ⊢ (¬ (𝜑 → ¬ 𝜓) → 𝜑)
7		rcp-NDASM2of2 194	. . . . 5 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜓) → ¬ 𝜓)
8	7	axL1-P3.21 252	. . . 4 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜓) → (𝜑 → ¬ 𝜓))
9		rcp-NDASM1of2 193	. . . 4 ⊢ ((¬ (𝜑 → ¬ 𝜓) ∧ ¬ 𝜓) → ¬ (𝜑 → ¬ 𝜓))
10	8, 9	rcp-NDNEGI2 219	. . 3 ⊢ (¬ (𝜑 → ¬ 𝜓) → ¬ ¬ 𝜓)
11	10	dnege-P3.30 276	. 2 ⊢ (¬ (𝜑 → ¬ 𝜓) → 𝜓)
12	6, 11	ndandi-P3.7 172	1 ⊢ (¬ (𝜑 → ¬ 𝜓) → (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132

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:  andasim-P3.46a  356