Theorem abssdv 3253

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 1885	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3248	. 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 1362   e. wcel 2164   {cab 2179   C_ wss 3153

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166

This theorem is referenced by:  fmpt  5708  tfrlemibacc  6379  tfrlemibfn  6381  tfr1onlembacc  6395  tfr1onlembfn  6397  tfrcllembacc  6408  tfrcllembfn  6410  eroprf  6682  genipv  7569  hashfacen  10907  4sqlemafi  12533  4sqlemffi  12534  4sqleminfi  12535  4sqlem11  12539  lss1d  13879  lspsn  13912  metrest  14674