Theorem mtbi 191

Description: An inference from a biconditional, related to modus tollens.

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	negbii 187	. 2 ⊢ (¬ φ ↔ ¬ ψ)
4	1, 3	mpbi 189	1 ⊢ ¬ ψ

Syntax hints:  ¬ wn 2   ↔ wb 146

This theorem is referenced by:  nvelv 2718  vnex 2720  opprc1b 2802  opthwiener 2813  dmsnsn0 3331  omsdomnn 4538  alephprc 4904  unialeph 4906  sinhalfpilem 8674

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147

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