Theorem twonotinotbothi 44316

Description: From these two negated implications it is not the case their nonnegated forms are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)

Hypotheses

Ref	Expression
twonotinotbothi.1	⊢ ¬ (𝜑 → 𝜓)
twonotinotbothi.2	⊢ ¬ (𝜒 → 𝜃)

Assertion

Ref	Expression
twonotinotbothi	⊢ ¬ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜃))

Proof of Theorem twonotinotbothi

Step	Hyp	Ref	Expression
1		twonotinotbothi.1	. . 3 ⊢ ¬ (𝜑 → 𝜓)
2	1	orci 861	. 2 ⊢ (¬ (𝜑 → 𝜓) ∨ ¬ (𝜒 → 𝜃))
3		pm3.14 992	. 2 ⊢ ((¬ (𝜑 → 𝜓) ∨ ¬ (𝜒 → 𝜃)) → ¬ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)))
4	2, 3	ax-mp 5	1 ⊢ ¬ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844

This theorem is referenced by: (None)