Theorem ssintrab 3950

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 2624	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
2	1	inteqi 3931	. . 3 |- |^|{x e. B | ph} = |^|{x | (x e. B /\ ph)}
3	2	sseq2i 3297	. 2 |- (A C_ |^|{x e. B | ph} <-> A C_ |^|{x | (x e. B /\ ph)})
4		impexp 433	. . . 4 |- (((x e. B /\ ph) -> A C_ x) <-> (x e. B -> (ph -> A C_ x)))
5	4	albii 1566	. . 3 |- (A.x((x e. B /\ ph) -> A C_ x) <-> A.x(x e. B -> (ph -> A C_ x)))
6		ssintab 3944	. . 3 |- (A C_ |^|{x | (x e. B /\ ph)} <-> A.x((x e. B /\ ph) -> A C_ x))
7		df-ral 2620	. . 3 |- (A.x e. B (ph -> A C_ x) <-> A.x(x e. B -> (ph -> A C_ x)))
8	5, 6, 7	3bitr4i 268	. 2 |- (A C_ |^|{x | (x e. B /\ ph)} <-> A.x e. B (ph -> A C_ x))
9	3, 8	bitri 240	1 |- (A C_ |^|{x e. B | ph} <-> A.x e. B (ph -> A C_ x))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619   C_ wss 3258  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by: (None)