Theorem ssintrab 3956

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 2520	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	inteqi 3937	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
3	2	sseq2i 3255	. 2 |- ( A C_ |^| { x e. B | ph } <-> A C_ |^| { x | ( x e. B /\ ph ) } )
4		impexp 263	. . . 4 |- ( ( ( x e. B /\ ph ) -> A C_ x ) <-> ( x e. B -> ( ph -> A C_ x ) ) )
5	4	albii 1519	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
6		ssintab 3950	. . 3 |- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A C_ x ) )
7		df-ral 2516	. . 3 |- ( A. x e. B ( ph -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
8	5, 6, 7	3bitr4i 212	. 2 |- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x e. B ( ph -> A C_ x ) )
9	3, 8	bitri 184	1 |- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   e. wcel 2202   {cab 2217   A.wral 2511   {crab 2515   C_ wss 3201   |^|cint 3933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-in 3207  df-ss 3214  df-int 3934

This theorem is referenced by: (None)