Theorem ssintab 4494

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 4493	. 2 |- ( A C_ |^| { x | ph } <-> A. y e. { x | ph } A C_ y )
2		sseq2 3627	. . 3 |- ( y = x -> ( A C_ y <-> A C_ x ) )
3	2	ralab2 3371	. 2 |- ( A. y e. { x | ph } A C_ y <-> A. x ( ph -> A C_ x ) )
4	1, 3	bitri 264	1 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   {cab 2608   A.wral 2912   C_ wss 3574   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  ssmin  4496  ssintrab  4500  intmin4  4506  dffi2  8329  rankval3b  8689  sstskm  9664  dfuzi  11468  cycsubg  17622  ssmclslem  31462  mptrcllem  37920  dfrcl2  37966  brtrclfv2  38019