Theorem abssdv 3221

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 1867	. 2 |- ( ph -> A. x ( ps -> x e. A ) )
3		abss 3216	. 2 |- ( { x | ps } C_ A <-> A. x ( ps -> x e. A ) )
4	2, 3	sylibr 133	1 |- ( ph -> { x | ps } C_ A )

Syntax hints:   -> wi 4   A.wal 1346   e. wcel 2141   {cab 2156   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134

This theorem is referenced by:  fmpt  5646  tfrlemibacc  6305  tfrlemibfn  6307  tfr1onlembacc  6321  tfr1onlembfn  6323  tfrcllembacc  6334  tfrcllembfn  6336  eroprf  6606  genipv  7471  hashfacen  10771  metrest  13300