Theorem sbabel 2255

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 1929	. . 3 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v [ y / x ] ( v = { z | ph } /\ v e. A ) )
2		sban 1878	. . . . 5 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) )
3		nfv 1467	. . . . . . . . . 10 |- F/ x z e. v
4	3	sbf 1708	. . . . . . . . 9 |- ( [ y / x ] z e. v <-> z e. v )
5	4	sbrbis 1884	. . . . . . . 8 |- ( [ y / x ] ( z e. v <-> ph ) <-> ( z e. v <-> [ y / x ] ph ) )
6	5	sbalv 1930	. . . . . . 7 |- ( [ y / x ] A. z ( z e. v <-> ph ) <-> A. z ( z e. v <-> [ y / x ] ph ) )
7		abeq2 2197	. . . . . . . 8 |- ( v = { z | ph } <-> A. z ( z e. v <-> ph ) )
8	7	sbbii 1696	. . . . . . 7 |- ( [ y / x ] v = { z | ph } <-> [ y / x ] A. z ( z e. v <-> ph ) )
9		abeq2 2197	. . . . . . 7 |- ( v = { z | [ y / x ] ph } <-> A. z ( z e. v <-> [ y / x ] ph ) )
10	6, 8, 9	3bitr4i 211	. . . . . 6 |- ( [ y / x ] v = { z | ph } <-> v = { z | [ y / x ] ph } )
11		sbabel.1	. . . . . . . 8 |- F/_ x A
12	11	nfcri 2223	. . . . . . 7 |- F/ x v e. A
13	12	sbf 1708	. . . . . 6 |- ( [ y / x ] v e. A <-> v e. A )
14	10, 13	anbi12i 449	. . . . 5 |- ( ( [ y / x ] v = { z | ph } /\ [ y / x ] v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
15	2, 14	bitri 183	. . . 4 |- ( [ y / x ] ( v = { z | ph } /\ v e. A ) <-> ( v = { z | [ y / x ] ph } /\ v e. A ) )
16	15	exbii 1542	. . 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 183	. 2 |- ( [ y / x ] E. v ( v = { z | ph } /\ v e. A ) <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
18		df-clel 2085	. . 3 |- ( { z | ph } e. A <-> E. v ( v = { z | ph } /\ v e. A ) )
19	18	sbbii 1696	. 2 |- ( [ y / x ] { z | ph } e. A <-> [ y / x ] E. v ( v = { z | ph } /\ v e. A ) )
20		df-clel 2085	. 2 |- ( { z | [ y / x ] ph } e. A <-> E. v ( v = { z | [ y / x ] ph } /\ v e. A ) )
21	17, 19, 20	3bitr4i 211	1 |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1288   = wceq 1290   E.wex 1427   e. wcel 1439   [wsb 1693   {cab 2075   F/_wnfc 2216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218

This theorem is referenced by: (None)