Theorem dmorgarev-L4.2 453

Description: De Morgan's Law A: Reverse Implication Lemma. †

Assertion

Ref	Expression
dmorgarev-L4.2	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))

Proof of Theorem dmorgarev-L4.2

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . 4 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → 𝜑)
2		rcp-NDASM1of3 195	. . . . 5 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → (¬ 𝜑 ∧ ¬ 𝜓))
3	2	ndander-P3.9 174	. . . 4 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → ¬ 𝜑)
4	1, 3	ndnege-P3.4 169	. . 3 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → ⊥)
5		rcp-NDASM3of3 197	. . . 4 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → 𝜓)
6		rcp-NDASM1of3 195	. . . . 5 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))
7	6	ndandel-P3.8 173	. . . 4 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → ¬ 𝜓)
8	5, 7	ndnege-P3.4 169	. . 3 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → ⊥)
9		rcp-NDASM2of2 194	. . 3 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓))
10	4, 8, 9	rcp-NDORE3 236	. 2 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∧ (𝜑 ∨ 𝜓)) → ⊥)
11	10	rcp-FALSENEGI2 434	1 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144  ⊥wff-false 157   ∧ 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-false-D2.5 158  df-rcp-AND3 161

This theorem is referenced by:  dmorga-P4.26a  456