Theorem nsyl3 627

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 626	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 615  ax-in2 616

This theorem is referenced by:  con2i  628  nsyl  629  pm2.65i  640  cesare  2146  cesaro  2150  pwnss  4189  sucprcreg  4582  reg3exmidlemwe  4612  reldmtpos  6308  snexxph  7011  elfi2  7033  ismkvnex  7216  fzn  10111  seq3f1olemqsum  10587  pcmpt2  12485  elply2  14914