Theorem elinti 3780

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elinti	|- ( A e. |^| B -> ( C e. B -> A e. C ) )

Proof of Theorem elinti

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 3779	. . 3 |- ( A e. |^| B -> ( A e. |^| B <-> A. x e. B A e. x ) )
2		eleq2 2203	. . . 4 |- ( x = C -> ( A e. x <-> A e. C ) )
3	2	rspccv 2786	. . 3 |- ( A. x e. B A e. x -> ( C e. B -> A e. C ) )
4	1, 3	syl6bi 162	. 2 |- ( A e. |^| B -> ( A e. |^| B -> ( C e. B -> A e. C ) ) )
5	4	pm2.43i 49	1 |- ( A e. |^| B -> ( C e. B -> A e. C ) )

Syntax hints:   -> wi 4   e. wcel 1480   A.wral 2416   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-int 3772

This theorem is referenced by: (None)