Theorem sylnbi 297

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

Hypotheses

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

Assertion

Ref	Expression
sylnbi	⊢ (¬ φ → χ)

Proof of Theorem sylnbi

Step	Hyp	Ref	Expression
1		sylnbi.1	. . 3 ⊢ (φ ↔ ψ)
2	1	notbii 287	. 2 ⊢ (¬ φ ↔ ¬ ψ)
3		sylnbi.2	. 2 ⊢ (¬ ψ → χ)
4	2, 3	sylbi 187	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:  sylnbir  298  reuun2  3539  iotanul  4355  ndmfv  5350  ndmovcom  5618  fvfullfun  5865