Theorem pm3.2ni 440

Description: Infer negated disjunction of negated premises.

Hypotheses

Ref	Expression
pm3.2ni.1	⊢ ¬ φ
pm3.2ni.2	⊢ ¬ ψ

Assertion

Ref	Expression
pm3.2ni	⊢ ¬ (φ ∨ ψ)

Proof of Theorem pm3.2ni

Step	Hyp	Ref	Expression
1		pm3.2ni.1	. . 3 ⊢ ¬ φ
2		pm3.2ni.2	. . 3 ⊢ ¬ ψ
3	1, 2	pm3.2i 234	. 2 ⊢ (¬ φ ∧ ¬ ψ)
4		ioran 254	. 2 ⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))
5	3, 4	mpbir 165	1 ⊢ ¬ (φ ∨ ψ)

Syntax hints:  ¬ wn 1   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  opprc1b 1906  tz7.44-2 2967  recgt0i 4385  halfnz 4586

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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