Theorem abssdv 3322

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 1927	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3317	. 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 1400   e. wcel 2209   {cab 2224   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233

This theorem is referenced by:  opabssxpd  4806  fmpt  5849  tfrlemibacc  6587  tfrlemibfn  6589  tfr1onlembacc  6603  tfr1onlembfn  6605  tfrcllembacc  6616  tfrcllembfn  6618  eroprf  6892  genipv  7866  hashfacen  11262  hashf1lem2  11264  4sqlemafi  13152  4sqlemffi  13153  4sqleminfi  13154  4sqlem11  13158  lss1d  14692  lspsn  14725  metrest  15530