Theorem vtoclgf 2345

Description: Implicit substitution of a class for a set variable, with bound-variable hypotheses in place of distinct variable restrictions. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
vtoclgf.1	|- (y e. A -> A.x y e. A)
vtoclgf.2	|- (ps -> A.xps)
vtoclgf.3	|- (x = A -> (ph <-> ps))
vtoclgf.4	|- ph

Assertion

Ref	Expression
vtoclgf	|- (A e. B -> ps)

Distinct variable group:   y,A

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		isset 2296	. . . 4 |- (A e. _V <-> E.z z = A)
3		vtoclgf.1	. . . . . . 7 |- (y e. A -> A.x y e. A)
4	3	hblem 1993	. . . . . 6 |- (z e. A -> A.x z e. A)
5	4	hbeleq 1997	. . . . 5 |- (z = A -> A.x z = A)
6		ax-17 1317	. . . . 5 |- (x = A -> A.z x = A)
7		eqeq1 1890	. . . . 5 |- (z = x -> (z = A <-> x = A))
8	5, 6, 7	cbvex 1529	. . . 4 |- (E.z z = A <-> E.x x = A)
9	2, 8	bitri 190	. . 3 |- (A e. _V <-> E.x x = A)
10		vtoclgf.2	. . . 4 |- (ps -> A.xps)
11		vtoclgf.4	. . . . 5 |- ph
12		vtoclgf.3	. . . . 5 |- (x = A -> (ph <-> ps))
13	11, 12	mpbii 210	. . . 4 |- (x = A -> ps)
14	10, 13	19.23ai 1412	. . 3 |- (E.x x = A -> ps)
15	9, 14	sylbi 216	. 2 |- (A e. _V -> ps)
16	1, 15	syl 12	1 |- (A e. B -> ps)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  vtoclg 2346  vtocl2gf 2348  vtoclgaf 2350  vtoclgafOLD 2351  ceqsexg 2392  elabgf 2404  ssiun2s 3297  reuuni2f 3810  reiota2f 5109

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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