Theorem sb8ab 2288

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 1953	. . 3 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		df-clab 2152	. . 3 |- ( z e. { y | [ y / x ] ph } <-> [ z / y ] [ y / x ] ph )
4		df-clab 2152	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
5	2, 3, 4	3bitr4ri 212	. 2 |- ( z e. { x | ph } <-> z e. { y | [ y / x ] ph } )
6	5	eqriv 2162	1 |- { x | ph } = { y | [ y / x ] ph }

Syntax hints:   = wceq 1343   F/wnf 1448   [wsb 1750   e. wcel 2136   {cab 2151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158

This theorem is referenced by: (None)