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  3491  undifexmid  4194  ordtriexmidlem2  4520  dmsn0el  5099  fidifsnen  6870  ctssdccl  7110  nninfwlpoimlemginf  7174  onntri35  7236  onntri45  7240  2omotaplemap  7256  exmidapne  7259  ltpopr  7594  caucvgprprlemnbj  7692  xrlttri3  9797  fzneuz  10101  iseqf1olemqcl  10486  iseqf1olemnab  10488  iseqf1olemab  10489  exp3val  10522  pwle2  14751