Theorem abssexg 2747

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 2746	. . 3 |- (A e. B -> P~A e. V)
2		rabexg 2724	. . 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 1652	. . 3 |- {x e. P~A | ph} = {x | (x e. P~A /\ ph)}
5		visset 1813	. . . . . 6 |- x e. V
6	5	elpw 2404	. . . . 5 |- (x e. P~A <-> x (_ A)
7	6	anbi1i 481	. . . 4 |- ((x e. P~A /\ ph) <-> (x (_ A /\ ph))
8	7	abbii 1575	. . 3 |- {x | (x e. P~A /\ ph)} = {x | (x (_ A /\ ph)}
9	4, 8	eqtr2 1496	. 2 |- {x | (x (_ A /\ ph)} = {x e. P~A | ph}
10	3, 9	syl5eqel 1552	1 |- (A e. B -> {x | (x (_ A /\ ph)} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  {cab 1463  {crab 1648  Vcvv 1811   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  pmex 4327  tgvalt 7616  tgval3t 7625  fctopOLD 7650  cctop 7652  cldval 7666  neif 7715  neival 7717  opnfval 7857  caufval 7926  issubg 8116  subsp 10554

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402

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