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  4622  funun  5362  pwuninel2  6434  swoer  6716  swoord1  6717  swoord2  6718  ssfirab  7106  djune  7253  exmidaclem  7398  sucpw1nss3  7428  onntri35  7430  onntri45  7434  elnnz  9464  lbioog  10117  ubioog  10118  fzneuz  10305  fzodisj  10384  fzodisjsn  10388  infssuzex  10461  fxnn0nninf  10669  zfz1isolemiso  11069  swrd0g  11200  infpnlem1  12890  exmidunben  13005  lgsdir2lem2  15716  2lgslem3  15788