Theorem trelded 38781

Description: Deduction form of trel 4759. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
trelded.1	|- ( ph -> Tr A )
trelded.2	|- ( ps -> B e. C )
trelded.3	|- ( ch -> C e. A )

Assertion

Ref	Expression
trelded	|- ( ( ph /\ ps /\ ch ) -> B e. A )

Proof of Theorem trelded

Step	Hyp	Ref	Expression
1		trelded.1	. 2 |- ( ph -> Tr A )
2		trelded.2	. 2 |- ( ps -> B e. C )
3		trelded.3	. 2 |- ( ch -> C e. A )
4		trel 4759	. . 3 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )
5	4	3impib 1262	. 2 |- ( ( Tr A /\ B e. C /\ C e. A ) -> B e. A )
6	1, 2, 3, 5	syl3an 1368	1 |- ( ( ph /\ ps /\ ch ) -> B e. A )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753

This theorem is referenced by:  suctrALT3  39160