Theorem ssintrab 3922

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 2495	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	inteqi 3903	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
3	2	sseq2i 3228	. 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 1494	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
6		ssintab 3916	. . 3 |- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A C_ x ) )
7		df-ral 2491	. . 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 1371   e. wcel 2178   {cab 2193   A.wral 2486   {crab 2490   C_ wss 3174   |^|cint 3899

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-in 3180  df-ss 3187  df-int 3900

This theorem is referenced by: (None)