Theorem bj-cleljustab 32847

Description: An instance of df-clel 2618 where the LHS (the definiendum) has the form "setvar e. class abstraction". The straightforward yet important fact that this statement can be proved from FOL= and df-clab 2609 (hence without df-clel 2618 or df-cleq 2615) was stressed by Mario Carneiro. The instance of df-clel 2618 where the LHS has the form "setvar e. setvar" is proved as cleljust 1998, from FOL= and ax-8 1992. Note: when df-ssb 32620 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2376 to prove bj-cleljustab 32847 from Tarski's FOL= with df-clab 2609. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cleljustab	|- ( x e. { y | ph } <-> E. z ( z = x /\ z e. { y | ph } ) )

Distinct variable groups:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-cleljustab

Step	Hyp	Ref	Expression
1		df-clab 2609	. 2 |- ( x e. { y | ph } <-> [ x / y ] ph )
2		ax6ev 1890	. . . 4 |- E. z z = x
3	2	biantrur 527	. . 3 |- ( [ x / y ] ph <-> ( E. z z = x /\ [ x / y ] ph ) )
4		19.41v 1914	. . . 4 |- ( E. z ( z = x /\ [ x / y ] ph ) <-> ( E. z z = x /\ [ x / y ] ph ) )
5	4	bicomi 214	. . 3 |- ( ( E. z z = x /\ [ x / y ] ph ) <-> E. z ( z = x /\ [ x / y ] ph ) )
6		sbequ 2376	. . . . . 6 |- ( x = z -> ( [ x / y ] ph <-> [ z / y ] ph ) )
7	6	equcoms 1947	. . . . 5 |- ( z = x -> ( [ x / y ] ph <-> [ z / y ] ph ) )
8	7	pm5.32i 669	. . . 4 |- ( ( z = x /\ [ x / y ] ph ) <-> ( z = x /\ [ z / y ] ph ) )
9	8	exbii 1774	. . 3 |- ( E. z ( z = x /\ [ x / y ] ph ) <-> E. z ( z = x /\ [ z / y ] ph ) )
10	3, 5, 9	3bitri 286	. 2 |- ( [ x / y ] ph <-> E. z ( z = x /\ [ z / y ] ph ) )
11		df-clab 2609	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
12	11	bicomi 214	. . . 4 |- ( [ z / y ] ph <-> z e. { y | ph } )
13	12	anbi2i 730	. . 3 |- ( ( z = x /\ [ z / y ] ph ) <-> ( z = x /\ z e. { y | ph } ) )
14	13	exbii 1774	. 2 |- ( E. z ( z = x /\ [ z / y ] ph ) <-> E. z ( z = x /\ z e. { y | ph } ) )
15	1, 10, 14	3bitri 286	1 |- ( x e. { y | ph } <-> E. z ( z = x /\ z e. { y | ph } ) )

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

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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609

This theorem is referenced by: (None)