Theorem mtbid 596

Description: A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)

Hypotheses

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

Assertion

Ref	Expression
mtbid	⊢ (φ → ¬ χ)

Proof of Theorem mtbid

Step	Hyp	Ref	Expression
1		mtbid.min	. 2 ⊢ (φ → ¬ ψ)
2		mtbid.maj	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	biimprd 147	. 2 ⊢ (φ → (χ → ψ))
4	1, 3	mtod 588	1 ⊢ (φ → ¬ χ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sylnib  600  eqneltrrd  2131  neleqtrd  2132  eueq3dc  2709  nqnq0pi  6421  zdclt  8094  frec2uzf1od  8873  expnegap0  8917