Theorem mtbi 289

Description: An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Hypotheses

Ref	Expression
mtbi.1	⊢ ¬ φ
mtbi.2	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
mtbi	⊢ ¬ ψ

Proof of Theorem mtbi

Step	Hyp	Ref	Expression
1		mtbi.1	. 2 ⊢ ¬ φ
2		mtbi.2	. . 3 ⊢ (φ ↔ ψ)
3	2	biimpri 197	. 2 ⊢ (ψ → φ)
4	1, 3	mto 167	1 ⊢ ¬ ψ

Syntax hints:  ¬ wn 3   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  mtbir  290  mtp-xorOLD  1537