Theorem inegd 1421

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 1420	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104  ⊥wfal 1407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is used by:  genpdisj  7896  cauappcvgprlemdisj  8024  caucvgprlemdisj  8047  caucvgprprlemdisj  8075  suplocexprlemdisj  8093  suplocexprlemub  8096  suplocsrlem  8181  resqrexlemgt0  11808  resqrexlemoverl  11809  leabs  11862  climge0  12116  isprm5lem  12945  ennnfonelemex  13363  dedekindeu  15849  dedekindicclemicc  15858  usgr1vr  16694  pw1nct  17238