Theorem mtbii 685

Description: An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)

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 158	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
4	1, 3	mtoi 674	1 ⊢ (𝜑 → ¬ 𝜒)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  onsucelsucexmid  4677  nntri2  6767  nntri3  6770  nndceq  6772  inffiexmid  7213  genpdisj  7890  ltposr  8130  hashennn  11219  fsumsplit  12174  sumsplitdc  12199  fprodm1  12365  m1dvdsndvds  13027  ballotfilemi1  13245  ballotfilemii  13246