Theorem sylnib 676

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 672	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 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  677  neqcomd  2182  inssdif0im  3490  undifexmid  4193  ordtriexmidlem2  4519  dmsn0el  5098  fidifsnen  6869  ctssdccl  7109  nninfwlpoimlemginf  7173  onntri35  7235  onntri45  7239  2omotaplemap  7255  exmidapne  7258  ltpopr  7593  caucvgprprlemnbj  7691  xrlttri3  9795  fzneuz  10098  iseqf1olemqcl  10483  iseqf1olemnab  10485  iseqf1olemab  10486  exp3val  10519  pwle2  14630