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  2239  inssdif0im  3580  undifexmid  4311  ordtriexmidlem2  4647  dmsn0el  5237  fidifsnen  7138  ctssdccl  7415  nninfwlpoimlemginf  7480  onntri35  7560  onntri45  7564  2omotaplemap  7587  exmidapne  7590  ltpopr  7926  caucvgprprlemnbj  8024  xrlttri3  10149  fzneuz  10457  iseqf1olemqcl  10885  iseqf1olemnab  10887  iseqf1olemab  10888  exp3val  10927  ballotfilemimin  13193  ballotfilemfrcn0  13217  pwle2  16884