Theorem ssintrab 3854

Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)

Assertion

Ref	Expression
ssintrab	|- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem ssintrab

Step	Hyp	Ref	Expression
1		df-rab 2457	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	inteqi 3835	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
3	2	sseq2i 3174	. 2 |- ( A C_ |^| { x e. B | ph } <-> A C_ |^| { x | ( x e. B /\ ph ) } )
4		impexp 261	. . . 4 |- ( ( ( x e. B /\ ph ) -> A C_ x ) <-> ( x e. B -> ( ph -> A C_ x ) ) )
5	4	albii 1463	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
6		ssintab 3848	. . 3 |- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A C_ x ) )
7		df-ral 2453	. . 3 |- ( A. x e. B ( ph -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
8	5, 6, 7	3bitr4i 211	. 2 |- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x e. B ( ph -> A C_ x ) )
9	3, 8	bitri 183	1 |- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   e. wcel 2141   {cab 2156   A.wral 2448   {crab 2452   C_ wss 3121   |^|cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by: (None)