Theorem sbabel 2399

Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
sbabel.1	|- F/_ x A

Assertion

Ref	Expression
sbabel	|- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   A( x, y, z)

Proof of Theorem sbabel

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbex 2055	. . 3 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v [ y / x ] ( v = { z | ph } /\ v e. A ) )
2		sban 2006	. . . . 5 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) )
3		nfv 1574	. . . . . . . . . 10 |- F/ x z e. v
4	3	sbf 1823	. . . . . . . . 9 |- ( [ y / x ] z e. v <-> z e. v )
5	4	sbrbis 2012	. . . . . . . 8 |- ( [ y / x ] ( z e. v <-> ph ) <-> ( z e. v <-> [ y / x ] ph ) )
6	5	sbalv 2056	. . . . . . 7 |- ( [ y / x ] A. z ( z e. v <-> ph ) <-> A. z ( z e. v <-> [ y / x ] ph ) )
7		abeq2 2338	. . . . . . . 8 |- ( v = { z | ph } <-> A. z ( z e. v <-> ph ) )
8	7	sbbii 1811	. . . . . . 7 |- ( [ y / x ] v = { z | ph } <-> [ y / x ] A. z ( z e. v <-> ph ) )
9		abeq2 2338	. . . . . . 7 |- ( v = { z | [ y / x ] ph } <-> A. z ( z e. v <-> [ y / x ] ph ) )
10	6, 8, 9	3bitr4i 212	. . . . . 6 |- ( [ y / x ] v = { z | ph } <-> v = { z | [ y / x ] ph } )
11		sbabel.1	. . . . . . . 8 |- F/_ x A
12	11	nfcri 2366	. . . . . . 7 |- F/ x v e. A
13	12	sbf 1823	. . . . . 6 |- ( [ y / x ] v e. A <-> v e. A )
14	10, 13	anbi12i 460	. . . . 5 |- ( ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
15	2, 14	bitri 184	. . . 4 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
16	15	exbii 1651	. . 3 |- ( E. v [ y / x ] ( v = { z | ph } /\ v e. A ) <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
17	1, 16	bitri 184	. 2 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
18		df-clel 2225	. . 3 |- ( { z | ph } e. A <-> E. v ( v = { z | ph } /\ v e. A ) )
19	18	sbbii 1811	. 2 |- ( [ y / x ] { z | ph } e. A <-> [ y / x ] E. v ( v = { z | ph } /\ v e. A ) )
20		df-clel 2225	. 2 |- ( { z | [ y / x ] ph } e. A <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
21	17, 19, 20	3bitr4i 212	1 |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   [wsb 1808   e. wcel 2200   {cab 2215   F/_wnfc 2359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by: (None)