Theorem abssexg 2753

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 2752	. . 3 |- (A e. B -> P~A e. V)
2		rabexg 2729	. . 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 1655	. . 3 |- {x e. P~A | ph} = {x | (x e. P~A /\ ph)}
5		visset 1816	. . . . . 6 |- x e. V
6	5	elpw 2408	. . . . 5 |- (x e. P~A <-> x (_ A)
7	6	anbi1i 483	. . . 4 |- ((x e. P~A /\ ph) <-> (x (_ A /\ ph))
8	7	abbii 1578	. . 3 |- {x | (x e. P~A /\ ph)} = {x | (x (_ A /\ ph)}
9	4, 8	eqtr2 1499	. 2 |- {x | (x (_ A /\ ph)} = {x e. P~A | ph}
10	3, 9	syl5eqel 1555	1 |- (A e. B -> {x | (x (_ A /\ ph)} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  {cab 1466  {crab 1651  Vcvv 1814   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  pmex 4333  tgvalt 7615  tgval3t 7624  fctopOLD 7647  cctop 7649  cldval 7663  neif 7712  neival 7714  opnfval 7854  caufval 7923  issubg 8112  subsp 10540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406

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