Theorem dmorgbrev-L4.4 455

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

Assertion

Ref	Expression
dmorgbrev-L4.4	⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Proof of Theorem dmorgbrev-L4.4

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . 3 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∧ ¬ 𝜑) → ¬ 𝜑)
2	1	nandir-P4.3b 375	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∧ ¬ 𝜑) → ¬ (𝜑 ∧ 𝜓))
3		rcp-NDASM2of2 194	. . 3 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∧ ¬ 𝜓) → ¬ 𝜓)
4	3	nandil-P4.3a 373	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∧ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
5		rcp-NDASM1of1 192	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
6	2, 4, 5	rcp-NDORE2 235	1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144

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

This theorem is referenced by:  dmorgb-P4.26b  457  dmorgbint-P4.26c  458