Theorem abssdv 3311

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Hypothesis

Ref	Expression
abssdv.1	|- ( ph -> ( ps -> x e. A ) )

Assertion

Ref	Expression
abssdv	|- ( ph -> { x | ps } C_ A )

Distinct variable groups:   ph, x   x, A

Allowed substitution hint:   ps( x)

Proof of Theorem abssdv

Step	Hyp	Ref	Expression
1		abssdv.1	. . 3 |- ( ph -> ( ps -> x e. A ) )
2	1	alrimiv 1923	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3306	. 2 |- ( { x | ps } C_ A <-> A. x ( ps -> x e. A ) )
4	2, 3	sylibr 134	1 |- ( ph -> { x | ps } C_ A )

Syntax hints:   -> wi 4   A.wal 1396   e. wcel 2203   {cab 2218   C_ wss 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3216  df-ss 3223

This theorem is referenced by:  opabssxpd  4785  fmpt  5826  tfrlemibacc  6556  tfrlemibfn  6558  tfr1onlembacc  6572  tfr1onlembfn  6574  tfrcllembacc  6585  tfrcllembfn  6587  eroprf  6861  genipv  7823  hashfacen  11204  4sqlemafi  13089  4sqlemffi  13090  4sqleminfi  13091  4sqlem11  13095  lss1d  14523  lspsn  14556  metrest  15363