Theorem sylnib 682

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 678	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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  683  neqcomd  2236  inssdif0im  3562  undifexmid  4283  ordtriexmidlem2  4618  dmsn0el  5206  fidifsnen  7057  ctssdccl  7310  nninfwlpoimlemginf  7375  onntri35  7455  onntri45  7459  2omotaplemap  7476  exmidapne  7479  ltpopr  7815  caucvgprprlemnbj  7913  xrlttri3  10032  fzneuz  10336  iseqf1olemqcl  10762  iseqf1olemnab  10764  iseqf1olemab  10765  exp3val  10804  pwle2  16625