Theorem sylnbi 689

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	|- ( ph <-> ps )
sylnbi.2	|- ( -. ps -> ch )

Assertion

Ref	Expression
sylnbi	|- ( -. ph -> ch )

Proof of Theorem sylnbi

Step	Hyp	Ref	Expression
1		sylnbi.1	. . 3 |- ( ph <-> ps )
2	1	notbii 678	. 2 |- ( -. ph <-> -. ps )
3		sylnbi.2	. 2 |- ( -. ps -> ch )
4	2, 3	sylbi 121	1 |- ( -. ph -> ch )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  sylnbir  690  mo2n  2114  reuun2  3516  regexmidlem1  4680  iotanul  5353  riotaund  6075  snnen2og  7160