Theorem sylnib 295

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

Hypotheses

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

Assertion

Ref	Expression
sylnib	⊢ (φ → ¬ χ)

Proof of Theorem sylnib

Step	Hyp	Ref	Expression
1		sylnib.1	. 2 ⊢ (φ → ¬ ψ)
2		sylnib.2	. . 3 ⊢ (ψ ↔ χ)
3	2	a1i 10	. 2 ⊢ (φ → (ψ ↔ χ))
4	1, 3	mtbid 291	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:  sylnibr  296  ssnelpss  3614  nnc3n3p1  6279  nchoicelem1  6290