Theorem sylnbi 678

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 668	. 2 |- ( -. ph <-> -. ps )
3		sylnbi.2	. 2 |- ( -. ps -> ch )
4	2, 3	sylbi 121	1 |- ( -. ph -> ch )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnbir  679  mo2n  2054  reuun2  3419  regexmidlem1  4533  iotanul  5194  riotaund  5865  snnen2og  6859