Theorem nsyl3 635

Description: A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)

Hypotheses

Ref	Expression
nsyl3.1	⊢ (𝜑 → ¬ 𝜓)
nsyl3.2	⊢ (𝜒 → 𝜓)

Assertion

Ref	Expression
nsyl3	⊢ (𝜒 → ¬ 𝜑)

Proof of Theorem nsyl3

Step	Hyp	Ref	Expression
1		nsyl3.2	. 2 ⊢ (𝜒 → 𝜓)
2		nsyl3.1	. . 3 ⊢ (𝜑 → ¬ 𝜓)
3	2	a1i 9	. 2 ⊢ (𝜒 → (𝜑 → ¬ 𝜓))
4	1, 3	mt2d 634	1 ⊢ (𝜒 → ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 623  ax-in2 624

This theorem is used by:  con2i  636  nsyl  637  pm2.65i  648  cesare  2191  cesaro  2195  pwnss  4298  sucprcreg  4698  reg3exmidlemwe  4728  reldmtpos  6529  snexxph  7274  elfi2  7311  ismkvnex  7501  fzn  10462  seq3f1olemqsum  10970  pcmpt2  13151  elply2  15978  chtqub  16313  umgredgnlp  16615  clwwlkn0  16871