Theorem sbabel 2793

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.) (Proof shortened by Wolf Lammen, 26-Dec-2019.)

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 2463	. . 3 |- ( [ y / x ] E. v ( v e. A /\ A. z ( z e. v <-> ph ) ) <-> E. v [ y / x ] ( v e. A /\ A. z ( z e. v <-> ph ) ) )
2		sban 2399	. . . . 5 |- ( [ y / x ] ( v e. A /\ A. z ( z e. v <-> ph ) ) <-> ( [ y / x ] v e. A /\ [ y / x ] A. z ( z e. v <-> ph ) ) )
3		sbabel.1	. . . . . . . 8 |- F/_ x A
4	3	nfcri 2758	. . . . . . 7 |- F/ x v e. A
5	4	sbf 2380	. . . . . 6 |- ( [ y / x ] v e. A <-> v e. A )
6		nfv 1843	. . . . . . . . 9 |- F/ x z e. v
7	6	sbf 2380	. . . . . . . 8 |- ( [ y / x ] z e. v <-> z e. v )
8	7	sbrbis 2405	. . . . . . 7 |- ( [ y / x ] ( z e. v <-> ph ) <-> ( z e. v <-> [ y / x ] ph ) )
9	8	sbalv 2464	. . . . . 6 |- ( [ y / x ] A. z ( z e. v <-> ph ) <-> A. z ( z e. v <-> [ y / x ] ph ) )
10	5, 9	anbi12i 733	. . . . 5 |- ( ( [ y / x ] v e. A /\ [ y / x ] A. z ( z e. v <-> ph ) ) <-> ( v e. A /\ A. z ( z e. v <-> [ y / x ] ph ) ) )
11	2, 10	bitri 264	. . . 4 |- ( [ y / x ] ( v e. A /\ A. z ( z e. v <-> ph ) ) <-> ( v e. A /\ A. z ( z e. v <-> [ y / x ] ph ) ) )
12	11	exbii 1774	. . 3 |- ( E. v [ y / x ] ( v e. A /\ A. z ( z e. v <-> ph ) ) <-> E. v ( v e. A /\ A. z ( z e. v <-> [ y / x ] ph ) ) )
13	1, 12	bitri 264	. 2 |- ( [ y / x ] E. v ( v e. A /\ A. z ( z e. v <-> ph ) ) <-> E. v ( v e. A /\ A. z ( z e. v <-> [ y / x ] ph ) ) )
14		clabel 2749	. . 3 |- ( { z | ph } e. A <-> E. v ( v e. A /\ A. z ( z e. v <-> ph ) ) )
15	14	sbbii 1887	. 2 |- ( [ y / x ] { z | ph } e. A <-> [ y / x ] E. v ( v e. A /\ A. z ( z e. v <-> ph ) ) )
16		clabel 2749	. 2 |- ( { z | [ y / x ] ph } e. A <-> E. v ( v e. A /\ A. z ( z e. v <-> [ y / x ] ph ) ) )
17	13, 15, 16	3bitr4i 292	1 |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)