Theorem sylbb2 138

Description: A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)

Hypotheses

Ref	Expression
sylbb2.1	|- ( ph <-> ps )
sylbb2.2	|- ( ch <-> ps )

Assertion

Ref	Expression
sylbb2	|- ( ph -> ch )

Proof of Theorem sylbb2

Step	Hyp	Ref	Expression
1		sylbb2.1	. 2 |- ( ph <-> ps )
2		sylbb2.2	. . 3 |- ( ch <-> ps )
3	2	biimpri 133	. 2 |- ( ps -> ch )
4	1, 3	sylbi 121	1 |- ( ph -> ch )

This proof depends on syntax axioms:   -> 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

This proof depends on definitions:  df-bi 117

This theorem is used by:  inffiexmid  7213  ssfirab  7244  ctssexmid  7490  pw1nel3  7590  fsumsplitsnun  12186  wlkm  16580  konigsberglem5  16733  wexmiddiffilem  17043  wexmiddifxylem  17045