Theorem ssintrab 3629

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 2309	. . . 4 |- {x e. B | ph } = {x | (x e. B /\ ph) }
2	1	inteqi 3610	. . 3 |- |^| {x e. B | ph } = |^| {x | (x e. B /\ ph) }
3	2	sseq2i 2964	. 2 |- (A C_ |^| {x e. B | ph } <-> A C_ |^| {x | (x e. B /\ ph) })
4		impexp 250	. . . 4 |- (((x e. B /\ ph) -> A C_ x) <-> (x e. B -> (ph -> A C_ x)))
5	4	albii 1356	. . 3 |- (A.x((x e. B /\ ph) -> A C_ x) <-> A.x(x e. B -> (ph -> A C_ x)))
6		ssintab 3623	. . 3 |- (A C_ |^| {x | (x e. B /\ ph) } <-> A.x((x e. B /\ ph) -> A C_ x))
7		df-ral 2305	. . 3 |- (A.x e. B (ph -> A C_ x) <-> A.x(x e. B -> (ph -> A C_ x)))
8	5, 6, 7	3bitr4i 201	. 2 |- (A C_ |^| {x | (x e. B /\ ph) } <-> A.x e. B (ph -> A C_ x))
9	3, 8	bitri 173	1 |- (A C_ |^| {x e. B | ph } <-> A.x e. B (ph -> A C_ x))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   e. wcel 1390   {cab 2023  A.wral 2300   {crab 2304   C_ wss 2911   |^|cint 3606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607

This theorem is referenced by: (None)