Theorem abssdv 3069

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 1796	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3064	. 2 |- ( { x | ps } C_ A <-> A. x ( ps -> x e. A ) )
4	2, 3	sylibr 132	1 |- ( ph -> { x | ps } C_ A )

Syntax hints:   -> wi 4   A.wal 1283   e. wcel 1434   {cab 2068   C_ wss 2974

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987

This theorem is referenced by:  fmpt  5345  tfrlemibacc  5969  tfrlemibfn  5971  tfr1onlembacc  5985  tfr1onlembfn  5987  tfrcllembacc  5998  tfrcllembfn  6000  eroprf  6258  genipv  6750