Theorem abssdv 3298

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 1920	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3293	. 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 1393   e. wcel 2200   {cab 2215   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210

This theorem is referenced by:  fmpt  5785  tfrlemibacc  6472  tfrlemibfn  6474  tfr1onlembacc  6488  tfr1onlembfn  6490  tfrcllembacc  6501  tfrcllembfn  6503  eroprf  6775  genipv  7696  hashfacen  11058  4sqlemafi  12918  4sqlemffi  12919  4sqleminfi  12920  4sqlem11  12924  lss1d  14347  lspsn  14380  metrest  15180