Theorem 3bitr4d 276

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)

Hypotheses

Ref	Expression
3bitr4d.1	|- (ph -> (ps <-> ch))
3bitr4d.2	|- (ph -> (th <-> ps))
3bitr4d.3	|- (ph -> (ta <-> ch))

Assertion

Ref	Expression
3bitr4d	|- (ph -> (th <-> ta))

Proof of Theorem 3bitr4d

Step	Hyp	Ref	Expression
1		3bitr4d.2	. 2 |- (ph -> (th <-> ps))
2		3bitr4d.1	. . 3 |- (ph -> (ps <-> ch))
3		3bitr4d.3	. . 3 |- (ph -> (ta <-> ch))
4	2, 3	bitr4d 247	. 2 |- (ph -> (ps <-> ta))
5	1, 4	bitrd 244	1 |- (ph -> (th <-> ta))

Syntax hints:   -> wi 4   <-> wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  sbcom  2089  sbcom2  2114  r19.12sn  3790  lefinlteq  4464  eqtfinrelk  4487  tfinlefin  4503  opbrop  4842  fvopab3g  5387  unpreima  5409  inpreima  5410  respreima  5411  fconst5  5456  isotr  5496  ncseqnc  6129