Theorem wl-clelv2-just 33379

Description: Show that the definition df-wl-clelv2 33380 is conservative. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-clelv2-just	|- ( x e. A <-> A. u ( u = x -> u e. A ) )

Distinct variable group:   x, u, A

Proof of Theorem wl-clelv2-just

Step	Hyp	Ref	Expression
1		ax-wl-8cl 33377	. . . 4 |- ( u = x -> ( u e. A -> x e. A ) )
2		ax-wl-8cl 33377	. . . . 5 |- ( x = u -> ( x e. A -> u e. A ) )
3	2	equcoms 1947	. . . 4 |- ( u = x -> ( x e. A -> u e. A ) )
4	1, 3	impbid 202	. . 3 |- ( u = x -> ( u e. A <-> x e. A ) )
5	4	equsalvw 1931	. 2 |- ( A. u ( u = x -> u e. A ) <-> x e. A )
6	5	bicomi 214	1 |- ( x e. A <-> A. u ( u = x -> u e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel-wl 33373

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-wl-8cl 33377

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)