Theorem sylan9bbr 681

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)

Hypotheses

Ref	Expression
sylan9bbr.1	|- (ph -> (ps <-> ch))
sylan9bbr.2	|- (th -> (ch <-> ta))

Assertion

Ref	Expression
sylan9bbr	|- ((th /\ ph) -> (ps <-> ta))

Proof of Theorem sylan9bbr

Step	Hyp	Ref	Expression
1		sylan9bbr.1	. . 3 |- (ph -> (ps <-> ch))
2		sylan9bbr.2	. . 3 |- (th -> (ch <-> ta))
3	1, 2	sylan9bb 680	. 2 |- ((ph /\ th) -> (ps <-> ta))
4	3	ancoms 439	1 |- ((th /\ ph) -> (ps <-> ta))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

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  df-an 360

This theorem is referenced by:  pm5.75  903  sbcom  2089  sbcom2  2114  2mo  2282  2eu6  2289  elssetkg  4270  fconstfv  5457  f1oiso  5500  mpteq12f  5656  mpt2eq123  5662  fmpt2x  5731  sbth  6207