Theorem sylnib 687

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 683	1 ⊢ (𝜑 → ¬ 𝜒)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  sylnibr  688  neqcomd  2243  inssdif0imOLD  3593  undifexmid  4330  ordtriexmidlem2  4667  dmsn0el  5257  fidifsnen  7172  ctssdccl  7452  nninfwlpoimlemginf  7517  onntri35  7597  onntri45  7601  2omotaplemap  7624  exmidapne  7627  ltpopr  7963  caucvgprprlemnbj  8061  xrlttri3  10210  fzneuz  10519  iseqf1olemqcl  10950  iseqf1olemnab  10952  iseqf1olemab  10953  exp3val  10992  ballotfilemimin  13300  ballotfilemfrcn0  13324  pwle2  17150  wexmiddiffilem  17165  wexmiddifxylem  17167