Theorem mtbird 292

Description: A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)

Hypotheses

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

Assertion

Ref	Expression
mtbird	⊢ (φ → ¬ ψ)

Proof of Theorem mtbird

Step	Hyp	Ref	Expression
1		mtbird.min	. 2 ⊢ (φ → ¬ χ)
2		mtbird.maj	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	biimpd 198	. 2 ⊢ (φ → (ψ → χ))
4	1, 3	mtod 168	1 ⊢ (φ → ¬ ψ)

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:  eqneltrd  2446  neleqtrrd  2449  ltfinirr  4458  nnadjoin  4521  vfin1cltv  4548  fvun1  5380  nnc3n3p1  6279  nnc3p1n3p2  6281  nchoicelem1  6290  nchoicelem2  6291  nchoicelem14  6303