Theorem andasim-P3.46-L1 354

Description: Lemma for andasim-P3.46a 356 and andasimint-P3.46b 357. †

Assertion

Ref	Expression
andasim-P3.46-L1	⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓))

Proof of Theorem andasim-P3.46-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 → ¬ 𝜓)) → (𝜑 ∧ 𝜓))
2	1	ndandel-P3.8 173	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 → ¬ 𝜓)) → 𝜓)
3	1	ndander-P3.9 174	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 → ¬ 𝜓)) → 𝜑)
4		rcp-NDASM2of2 194	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 → ¬ 𝜓)) → (𝜑 → ¬ 𝜓))
5	3, 4	ndime-P3.6 171	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜓)
6	2, 5	rcp-NDNEGI2 219	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-true-D2.4 155

This theorem is referenced by:  andasim-P3.46a  356  andasimint-P3.46b  357