Theorem sylnibr 296

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

Hypotheses

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

Assertion

Ref	Expression
sylnibr	⊢ (φ → ¬ χ)

Proof of Theorem sylnibr

Step	Hyp	Ref	Expression
1		sylnibr.1	. 2 ⊢ (φ → ¬ ψ)
2		sylnibr.2	. . 3 ⊢ (χ ↔ ψ)
3	2	bicomi 193	. 2 ⊢ (ψ ↔ χ)
4	1, 3	sylnib 295	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:  ncfinraise  4482  tfinltfin  4502  sfindbl  4531  tfinnn  4535  vfinncvntsp  4550  nnc3n3p1  6279  nnc3n3p2  6280  nnc3p1n3p2  6281  nchoicelem2  6291