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  7521  cauappcvgprlemdisj  7649  caucvgprlemdisj  7672  caucvgprprlemdisj  7700  suplocexprlemdisj  7718  suplocexprlemub  7721  suplocsrlem  7806  resqrexlemgt0  11028  resqrexlemoverl  11029  leabs  11082  climge0  11332  isprm5lem  12140  ennnfonelemex  12414  dedekindeu  14071  dedekindicclemicc  14080  pw1nct  14722