Theorem ssintab 3788

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 3787	. 2 |- ( A C_ |^| { x | ph } <-> A. y e. { x | ph } A C_ y )
2		sseq2 3121	. . 3 |- ( y = x -> ( A C_ y <-> A C_ x ) )
3	2	ralab2 2848	. 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 1329   {cab 2125   A.wral 2416   C_ wss 3071   |^|cint 3771

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-int 3772

This theorem is referenced by:  ssmin  3790  ssintrab  3794  intmin4  3799  dfuzi  9164