Theorem dmorgbfwd-L4.3 454

Description: De Morgan's Law B: Forward Implication Lemma.

Assertion

Ref	Expression
dmorgbfwd-L4.3	⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))

Proof of Theorem dmorgbfwd-L4.3

Step	Hyp	Ref	Expression
1		norer-P4.2b.CL 372	. . . 4 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → ¬ ¬ 𝜑)
2	1	dnege-P3.30 276	. . 3 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → 𝜑)
3		norel-P4.2a.CL 369	. . . 4 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → ¬ ¬ 𝜓)
4	3	dnege-P3.30 276	. . 3 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → 𝜓)
5	2, 4	ndandi-P3.7 172	. 2 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))
6	5	trnsp-P3.31b.RC 283	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  df-rcp-AND3 161

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