Theorem mtbi 681

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 133	. 2 ⊢ (𝜓 → 𝜑)
4	1, 3	mto 672	1 ⊢ ¬ 𝜓

Syntax hints:  ¬ wn 3   ↔ wb 105

This theorem was proved from 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 theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mtbir  682  vnex  4262  onsucelsucexmid  4675  dtruex  4704  dmsn0  5253  php5  7153  exmidonfinlem  7539  ndvdsi  12683  nprmi  12885  dec2dvds  13173  dec5dvds2  13175  ballotfilem2  13211  unennn  13271  bj-vprc  16905  bj-vnex  16907  trirec0xor  17068