Theorem abssexg 2737

Description: Existence of a class of subsets.

Assertion

Ref	Expression
abssexg	|- (A e. B -> {x | (x (_ A /\ ph)} e. V)

Distinct variable group:   x,A

Proof of Theorem abssexg

Step	Hyp	Ref	Expression
1		pwexg 2736	. . 3 |- (A e. B -> P~A e. V)
2		rabexg 2714	. . 3 |- (P~A e. V -> {x e. P~A | ph} e. V)
3	1, 2	syl 10	. 2 |- (A e. B -> {x e. P~A | ph} e. V)
4		df-rab 1644	. . 3 |- {x e. P~A | ph} = {x | (x e. P~A /\ ph)}
5		visset 1804	. . . . . 6 |- x e. V
6	5	elpw 2394	. . . . 5 |- (x e. P~A <-> x (_ A)
7	6	anbi1i 480	. . . 4 |- ((x e. P~A /\ ph) <-> (x (_ A /\ ph))
8	7	abbii 1567	. . 3 |- {x | (x e. P~A /\ ph)} = {x | (x (_ A /\ ph)}
9	4, 8	eqtr2 1488	. 2 |- {x | (x (_ A /\ ph)} = {x e. P~A | ph}
10	3, 9	syl5eqel 1544	1 |- (A e. B -> {x | (x (_ A /\ ph)} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  {cab 1456  {crab 1640  Vcvv 1802   (_ wss 2037  P~cpw 2391

This theorem is referenced by:  pmex 4311  tgvalt 7558  tgval3t 7567  fctop 7592  cctop 7594  cldval 7608  neif 7656  neival 7658  opnfval 7797  caufval 7864  issubg 8053  subsp 10429

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644  df-v 1803  df-in 2041  df-ss 2043  df-pw 2392

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