Theorem inegd 1351

Description: Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis

Ref	Expression
inegd.1	⊢ ((𝜑 ∧ 𝜓) → ⊥)

Assertion

Ref	Expression
inegd	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem inegd

Step	Hyp	Ref	Expression
1		inegd.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ⊥)
2	1	ex 114	. 2 ⊢ (𝜑 → (𝜓 → ⊥))
3		dfnot 1350	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
4	2, 3	sylibr 133	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ⊥wfal 1337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338

This theorem is referenced by:  genpdisj  7355  cauappcvgprlemdisj  7483  caucvgprlemdisj  7506  caucvgprprlemdisj  7534  suplocexprlemdisj  7552  suplocexprlemub  7555  suplocsrlem  7640  resqrexlemgt0  10824  resqrexlemoverl  10825  leabs  10878  climge0  11126  ennnfonelemex  11963  dedekindeu  12809  dedekindicclemicc  12818  pw1nct  13371