Theorem wl-ax8clv2 33381

Description: Axiom ax-wl-8cl 33377 carries over to our new definition df-wl-clelv2 33380. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-ax8clv2	|- ( x = y -> ( x e. A -> y e. A ) )

Proof of Theorem wl-ax8clv2

Dummy variables u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 1959	. 2 |- ( x = y <-> E. u ( u = x /\ u = y ) )
2		df-wl-clelv2 33380	. . . . 5 |- ( x e. A <-> A. t ( t = x -> t e. A ) )
3		equtrr 1949	. . . . . . 7 |- ( u = x -> ( t = u -> t = x ) )
4	3	imim1d 82	. . . . . 6 |- ( u = x -> ( ( t = x -> t e. A ) -> ( t = u -> t e. A ) ) )
5	4	alimdv 1845	. . . . 5 |- ( u = x -> ( A. t ( t = x -> t e. A ) -> A. t ( t = u -> t e. A ) ) )
6	2, 5	syl5bi 232	. . . 4 |- ( u = x -> ( x e. A -> A. t ( t = u -> t e. A ) ) )
7		equeuclr 1950	. . . . . . 7 |- ( u = y -> ( t = y -> t = u ) )
8	7	imim1d 82	. . . . . 6 |- ( u = y -> ( ( t = u -> t e. A ) -> ( t = y -> t e. A ) ) )
9	8	alimdv 1845	. . . . 5 |- ( u = y -> ( A. t ( t = u -> t e. A ) -> A. t ( t = y -> t e. A ) ) )
10		df-wl-clelv2 33380	. . . . 5 |- ( y e. A <-> A. t ( t = y -> t e. A ) )
11	9, 10	syl6ibr 242	. . . 4 |- ( u = y -> ( A. t ( t = u -> t e. A ) -> y e. A ) )
12	6, 11	sylan9 689	. . 3 |- ( ( u = x /\ u = y ) -> ( x e. A -> y e. A ) )
13	12	exlimiv 1858	. 2 |- ( E. u ( u = x /\ u = y ) -> ( x e. A -> y e. A ) )
14	1, 13	sylbi 207	1 |- ( x = y -> ( x e. A -> y e. A ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel-wl 33373   e. wcel2-wl 33375

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-wl-clelv2 33380

This theorem is referenced by: (None)