Theorem sb8ab 2200

Description: Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)

Hypothesis

Ref	Expression
sb8ab.1	|- F/ y ph

Assertion

Ref	Expression
sb8ab	|- { x | ph } = { y | [ y / x ] ph }

Proof of Theorem sb8ab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb8ab.1	. . . 4 |- F/ y ph
2	1	sbco2 1880	. . 3 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		df-clab 2068	. . 3 |- ( z e. { y | [ y / x ] ph } <-> [ z / y ] [ y / x ] ph )
4		df-clab 2068	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
5	2, 3, 4	3bitr4ri 211	. 2 |- ( z e. { x | ph } <-> z e. { y | [ y / x ] ph } )
6	5	eqriv 2078	1 |- { x | ph } = { y | [ y / x ] ph }

Syntax hints:   = wceq 1284   F/wnf 1389   e. wcel 1433   [wsb 1685   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074

This theorem is referenced by: (None)