Theorem mtbii 538

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

Hypotheses

Ref	Expression
mtbii.min	⊢ ¬ ψ
mtbii.maj	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
mtbii	⊢ (φ → ¬ χ)

Proof of Theorem mtbii

Step	Hyp	Ref	Expression
1		mtbii.min	. 2 ⊢ ¬ ψ
2		mtbii.maj	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	biimprd 136	. 2 ⊢ (φ → (χ → ψ))
4	1, 3	mtoi 94	1 ⊢ (φ → ¬ χ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127

This theorem is referenced by:  ssnpss 1573  noel 1711  aceq6b 3565  nd3 3734  axunndlem1 3741  axregndlem1 3748  axregndlem2 3749  axregnd 3750  axacndlem5 3757  addnidpi 3822  indpi 3828  lt2sq 4414

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

This theorem depends on definitions:  df-bi 128

metamath.org