Theorem sb8ab 2362

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 2025	. . 3 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		df-clab 2225	. . 3 |- ( z e. { y | [ y / x ] ph } <-> [ z / y ] [ y / x ] ph )
4		df-clab 2225	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
5	2, 3, 4	3bitr4ri 213	. 2 |- ( z e. { x | ph } <-> z e. { y | [ y / x ] ph } )
6	5	eqriv 2235	1 |- { x | ph } = { y | [ y / x ] ph }

Syntax hints:   = wceq 1402   F/wnf 1513   [wsb 1815   e. wcel 2209   {cab 2224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231

This theorem is referenced by: (None)