Theorem nsyl 631

Description: A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)

Hypotheses

Ref	Expression
nsyl.1	⊢ (𝜑 → ¬ 𝜓)
nsyl.2	⊢ (𝜒 → 𝜓)

Assertion

Ref	Expression
nsyl	⊢ (𝜑 → ¬ 𝜒)

Proof of Theorem nsyl

Step	Hyp	Ref	Expression
1		nsyl.1	. . 3 ⊢ (𝜑 → ¬ 𝜓)
2		nsyl.2	. . 3 ⊢ (𝜒 → 𝜓)
3	1, 2	nsyl3 629	. 2 ⊢ (𝜒 → ¬ 𝜑)
4	3	con2i 630	1 ⊢ (𝜑 → ¬ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 617  ax-in2 618

This theorem is referenced by:  con3i  635  pm4.52im  755  intnand  936  intnanrd  937  intn3an1d  1390  intn3an2d  1391  intn3an3d  1392  camestres  2183  camestros  2187  calemes  2194  calemos  2197  unssin  3443  inssun  3444  onsucelsucexmid  4619  funun  5358  pwuninel2  6418  swoer  6698  swoord1  6699  swoord2  6700  ssfirab  7086  djune  7233  exmidaclem  7378  sucpw1nss3  7408  onntri35  7410  onntri45  7414  elnnz  9444  lbioog  10097  ubioog  10098  fzneuz  10285  fzodisj  10364  fzodisjsn  10368  infssuzex  10440  fxnn0nninf  10648  zfz1isolemiso  11048  swrd0g  11178  infpnlem1  12868  exmidunben  12983  lgsdir2lem2  15693  2lgslem3  15765