Theorem imbi13 38726

Description: Join three logical equivalences to form equivalence of implications. imbi13 38726 is imbi13VD 39110 without virtual deductions and was automatically derived from imbi13VD 39110 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
imbi13	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

Proof of Theorem imbi13

Step	Hyp	Ref	Expression
1		imbi12 336	. 2 |- ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ch -> ta ) <-> ( th -> et ) ) ) )
2		imbi12 336	. 2 |- ( ( ph <-> ps ) -> ( ( ( ch -> ta ) <-> ( th -> et ) ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) )
3	1, 2	syl9r 78	1 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196

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 197

This theorem is referenced by:  trsbc  38750  trsbcVD  39113