Theorem sylnib 642

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 638	1 ⊢ (𝜑 → ¬ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  sylnibr  643  inssdif0im  3377  undifexmid  4057  ordtriexmidlem2  4374  dmsn0el  4944  fidifsnen  6693  ctssdclemr  6911  ltpopr  7304  caucvgprprlemnbj  7402  xrlttri3  9424  fzneuz  9722  iseqf1olemqcl  10100  iseqf1olemnab  10102  iseqf1olemab  10103  exp3val  10136  pwle2  12779