Theorem sb8ab 2299

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 1965	. . 3 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		df-clab 2164	. . 3 |- ( z e. { y | [ y / x ] ph } <-> [ z / y ] [ y / x ] ph )
4		df-clab 2164	. . 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 2174	1 |- { x | ph } = { y | [ y / x ] ph }

Syntax hints:   = wceq 1353   F/wnf 1460   [wsb 1762   e. wcel 2148   {cab 2163

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170

This theorem is referenced by: (None)