Theorem ssintab 3841

Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ssintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3840	. 2 |- ( A C_ |^| { x | ph } <-> A. y e. { x | ph } A C_ y )
2		sseq2 3166	. . 3 |- ( y = x -> ( A C_ y <-> A C_ x ) )
3	2	ralab2 2890	. 2 |- ( A. y e. { x | ph } A C_ y <-> A. x ( ph -> A C_ x ) )
4	1, 3	bitri 183	1 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   {cab 2151   A.wral 2444   C_ wss 3116   |^|cint 3824

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  ssmin  3843  ssintrab  3847  intmin4  3852  dfuzi  9301