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  7451  nninfwlpoimlemginf  7516  onntri35  7596  onntri45  7600  2omotaplemap  7623  exmidapne  7626  ltpopr  7962  caucvgprprlemnbj  8060  xrlttri3  10201  fzneuz  10510  iseqf1olemqcl  10938  iseqf1olemnab  10940  iseqf1olemab  10941  exp3val  10980  ballotfilemimin  13251  ballotfilemfrcn0  13275  pwle2  17040  wexmiddiffilem  17055  wexmiddifxylem  17057