Theorem wl-ax8clv1 33378

Description: Lifting the distinct variable constraint on x and y in ax-wl-8cl 33377. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-ax8clv1	|- ( x = y -> ( x e. A -> y e. A ) )

Distinct variable groups:   x, A   y, A

Proof of Theorem wl-ax8clv1

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 1959	. 2 |- ( x = y <-> E. u ( u = x /\ u = y ) )
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		ax-wl-8cl 33377	. . . 4 |- ( u = y -> ( u e. A -> y e. A ) )
5	3, 4	sylan9 689	. . 3 |- ( ( u = x /\ u = y ) -> ( x e. A -> y e. A ) )
6	5	exlimiv 1858	. 2 |- ( E. u ( u = x /\ u = y ) -> ( x e. A -> y e. A ) )
7	1, 6	sylbi 207	1 |- ( x = y -> ( x e. A -> y e. A ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   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)