Theorem ssintab 3945

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 3944	. 2 |- ( A C_ |^| { x | ph } <-> A. y e. { x | ph } A C_ y )
2		sseq2 3251	. . 3 |- ( y = x -> ( A C_ y <-> A C_ x ) )
3	2	ralab2 2970	. 2 |- ( A. y e. { x | ph } A C_ y <-> A. x ( ph -> A C_ x ) )
4	1, 3	bitri 184	1 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   {cab 2217   A.wral 2510   C_ wss 3200   |^|cint 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-int 3929

This theorem is referenced by:  ssmin  3947  ssintrab  3951  intmin4  3956  dfuzi  9589