Theorem wl-ax11-lem2 33363

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem2	|- ( ( A. u u = y /\ -. A. x x = y ) -> A. x u = y )

Distinct variable group:   x, u

Proof of Theorem wl-ax11-lem2

Step	Hyp	Ref	Expression
1		sp 2053	. . 3 |- ( A. u u = y -> u = y )
2		aev 1983	. . . 4 |- ( A. x x = u -> A. x x = y )
3		pm2.21 120	. . . 4 |- ( -. A. x x = y -> ( A. x x = y -> A. x x = u ) )
4	2, 3	impbid2 216	. . 3 |- ( -. A. x x = y -> ( A. x x = u <-> A. x x = y ) )
5	1, 4	anim12i 590	. 2 |- ( ( A. u u = y /\ -. A. x x = y ) -> ( u = y /\ ( A. x x = u <-> A. x x = y ) ) )
6		wl-aleq 33322	. 2 |- ( A. x u = y <-> ( u = y /\ ( A. x x = u <-> A. x x = y ) ) )
7	5, 6	sylibr 224	1 |- ( ( A. u u = y /\ -. A. x x = y ) -> A. x u = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481

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-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  wl-ax11-lem3  33364