Theorem sylnbir 298

Description: A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)

Hypotheses

Ref	Expression
sylnbir.1	⊢ (ψ ↔ φ)
sylnbir.2	⊢ (¬ ψ → χ)

Assertion

Ref	Expression
sylnbir	⊢ (¬ φ → χ)

Proof of Theorem sylnbir

Step	Hyp	Ref	Expression
1		sylnbir.1	. . 3 ⊢ (ψ ↔ φ)
2	1	bicomi 193	. 2 ⊢ (φ ↔ ψ)
3		sylnbir.2	. 2 ⊢ (¬ ψ → χ)
4	2, 3	sylnbi 297	1 ⊢ (¬ φ → χ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  f0cli  5419  ndmov  5616  elovex12  5649  fvmptex  5722  nchoicelem18  6307