Theorem abssdv 3267

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 1897	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3262	. 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 1371   e. wcel 2176   {cab 2191   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179

This theorem is referenced by:  fmpt  5730  tfrlemibacc  6412  tfrlemibfn  6414  tfr1onlembacc  6428  tfr1onlembfn  6430  tfrcllembacc  6441  tfrcllembfn  6443  eroprf  6715  genipv  7622  hashfacen  10981  4sqlemafi  12718  4sqlemffi  12719  4sqleminfi  12720  4sqlem11  12724  lss1d  14145  lspsn  14178  metrest  14978