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  7320  onntri45  7324  2omotaplemap  7340  exmidapne  7343  ltpopr  7679  caucvgprprlemnbj  7777  xrlttri3  9889  fzneuz  10193  iseqf1olemqcl  10608  iseqf1olemnab  10610  iseqf1olemab  10611  exp3val  10650  pwle2  15729