Theorem nsyl3 631

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 630	1 ⊢ (𝜒 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 619  ax-in2 620

This theorem is referenced by:  con2i  632  nsyl  633  pm2.65i  644  cesare  2184  cesaro  2188  pwnss  4255  sucprcreg  4653  reg3exmidlemwe  4683  reldmtpos  6462  snexxph  7192  elfi2  7231  ismkvnex  7414  fzn  10339  seq3f1olemqsum  10838  pcmpt2  12997  elply2  15546  umgredgnlp  16093  clwwlkn0  16349