Theorem sbabel 2203

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 1880	. . 3 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v [ y / x ] ( v = { z | ph } /\ v e. A ) )
2		sban 1829	. . . . 5 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) )
3		nfv 1421	. . . . . . . . . 10 |- F/ x z e. v
4	3	sbf 1660	. . . . . . . . 9 |- ( [ y / x ] z e. v <-> z e. v )
5	4	sbrbis 1835	. . . . . . . 8 |- ( [ y / x ] ( z e. v <-> ph ) <-> ( z e. v <-> [ y / x ] ph ) )
6	5	sbalv 1881	. . . . . . 7 |- ( [ y / x ] A. z ( z e. v <-> ph ) <-> A. z ( z e. v <-> [ y / x ] ph ) )
7		abeq2 2146	. . . . . . . 8 |- ( v = { z | ph } <-> A. z ( z e. v <-> ph ) )
8	7	sbbii 1648	. . . . . . 7 |- ( [ y / x ] v = { z | ph } <-> [ y / x ] A. z ( z e. v <-> ph ) )
9		abeq2 2146	. . . . . . 7 |- ( v = { z | [ y / x ] ph } <-> A. z ( z e. v <-> [ y / x ] ph ) )
10	6, 8, 9	3bitr4i 201	. . . . . 6 |- ( [ y / x ] v = { z | ph } <-> v = { z | [ y / x ] ph } )
11		sbabel.1	. . . . . . . 8 |- F/_ x A
12	11	nfcri 2172	. . . . . . 7 |- F/ x v e. A
13	12	sbf 1660	. . . . . 6 |- ( [ y / x ] v e. A <-> v e. A )
14	10, 13	anbi12i 433	. . . . 5 |- ( ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
15	2, 14	bitri 173	. . . 4 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
16	15	exbii 1496	. . 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 173	. 2 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
18		df-clel 2036	. . 3 |- ( { z | ph } e. A <-> E. v ( v = { z | ph } /\ v e. A ) )
19	18	sbbii 1648	. 2 |- ( [ y / x ] { z | ph } e. A <-> [ y / x ] E. v ( v = { z | ph } /\ v e. A ) )
20		df-clel 2036	. 2 |- ( { z | [ y / x ] ph } e. A <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
21	17, 19, 20	3bitr4i 201	1 |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   [wsb 1645   {cab 2026   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by: (None)