Theorem ssintab 3824

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 3823	. 2 |- ( A C_ |^| { x | ph } <-> A. y e. { x | ph } A C_ y )
2		sseq2 3152	. . 3 |- ( y = x -> ( A C_ y <-> A C_ x ) )
3	2	ralab2 2876	. 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 1333   {cab 2143   A.wral 2435   C_ wss 3102   |^|cint 3807

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3808

This theorem is referenced by:  ssmin  3826  ssintrab  3830  intmin4  3835  dfuzi  9257