Theorem nsyl3 626

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 625	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 614  ax-in2 615

This theorem is referenced by:  con2i  627  nsyl  628  pm2.65i  639  cesare  2130  cesaro  2134  pwnss  4161  sucprcreg  4550  reg3exmidlemwe  4580  reldmtpos  6256  snexxph  6951  elfi2  6973  ismkvnex  7155  fzn  10044  seq3f1olemqsum  10502  pcmpt2  12344