Theorem mtbid 291

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 214	. 2 ⊢ (φ → (χ → ψ))
4	1, 3	mtod 168	1 ⊢ (φ → ¬ χ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  sylnib  295  eqneltrrd  2447  neleqtrd  2448  eueq3  3012  nnadjoinpw  4522  nnc3n3p2  6280