Theorem dmorgafwd-L4.1 452

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

Assertion

Ref	Expression
dmorgafwd-L4.1	⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))

Proof of Theorem dmorgafwd-L4.1

Step	Hyp	Ref	Expression
1		norer-P4.2b.CL 372	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		norel-P4.2a.CL 369	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
3	1, 2	ndandi-P3.7 172	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:  dmorga-P4.26a  456