Theorem sylnib 683

Description: A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)

Hypotheses

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

Assertion

Ref	Expression
sylnib	⊢ (𝜑 → ¬ 𝜒)

Proof of Theorem sylnib

Step	Hyp	Ref	Expression
1		sylnib.1	. 2 ⊢ (𝜑 → ¬ 𝜓)
2		sylnib.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
3	2	a1i 9	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 3	mtbid 679	1 ⊢ (𝜑 → ¬ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  684  neqcomd  2237  inssdif0im  3575  undifexmid  4305  ordtriexmidlem2  4641  dmsn0el  5231  fidifsnen  7124  ctssdccl  7401  nninfwlpoimlemginf  7466  onntri35  7546  onntri45  7550  2omotaplemap  7570  exmidapne  7573  ltpopr  7909  caucvgprprlemnbj  8007  xrlttri3  10129  fzneuz  10434  iseqf1olemqcl  10860  iseqf1olemnab  10862  iseqf1olemab  10863  exp3val  10902  pwle2  16764