Theorem sylnib 677

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 673	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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  678  neqcomd  2201  inssdif0im  3519  undifexmid  4227  ordtriexmidlem2  4557  dmsn0el  5140  fidifsnen  6940  ctssdccl  7186  nninfwlpoimlemginf  7251  onntri35  7322  onntri45  7326  2omotaplemap  7342  exmidapne  7345  ltpopr  7681  caucvgprprlemnbj  7779  xrlttri3  9891  fzneuz  10195  iseqf1olemqcl  10610  iseqf1olemnab  10612  iseqf1olemab  10613  exp3val  10652  pwle2  15753