Theorem inegd 1383

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 115	. 2 ⊢ (𝜑 → (𝜓 → ⊥))
3		dfnot 1382	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ⊥wfal 1369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by:  genpdisj  7585  cauappcvgprlemdisj  7713  caucvgprlemdisj  7736  caucvgprprlemdisj  7764  suplocexprlemdisj  7782  suplocexprlemub  7785  suplocsrlem  7870  resqrexlemgt0  11167  resqrexlemoverl  11168  leabs  11221  climge0  11471  isprm5lem  12282  ennnfonelemex  12574  dedekindeu  14802  dedekindicclemicc  14811  pw1nct  15563