Theorem inegd 1372

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 1371	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ⊥wfal 1358

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 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359

This theorem is referenced by:  genpdisj  7517  cauappcvgprlemdisj  7645  caucvgprlemdisj  7668  caucvgprprlemdisj  7696  suplocexprlemdisj  7714  suplocexprlemub  7717  suplocsrlem  7802  resqrexlemgt0  11020  resqrexlemoverl  11021  leabs  11074  climge0  11324  isprm5lem  12131  ennnfonelemex  12405  dedekindeu  13883  dedekindicclemicc  13892  pw1nct  14523