Theorem wl-dveeq12 33311

Description: The current form of ax-13 2246 has a particular disadvantage: The condition -. x = y is less versatile than the general form -. A. x x = y. You need ax-10 2019 to arrive at the more general form presented here. You need 19.8a 2052 (or ax-12 2047) to restore y = z from E. x y = z again. (Contributed by Wolf Lammen, 9-Jun-2021.)

Assertion

Ref	Expression
wl-dveeq12	|- ( -. A. x x = y -> ( E. x z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem wl-dveeq12

Step	Hyp	Ref	Expression
1		exnal 1754	. 2 |- ( E. x -. x = y <-> -. A. x x = y )
2		hbe1 2021	. . 3 |- ( E. x z = y -> A. x E. x z = y )
3		ax13lem2 2296	. . . . . 6 |- ( -. x = y -> ( E. x z = y -> z = y ) )
4		ax13lem1 2248	. . . . . 6 |- ( -. x = y -> ( z = y -> A. x z = y ) )
5	3, 4	syldc 48	. . . . 5 |- ( E. x z = y -> ( -. x = y -> A. x z = y ) )
6	5	aleximi 1759	. . . 4 |- ( A. x E. x z = y -> ( E. x -. x = y -> E. x A. x z = y ) )
7	6	com12 32	. . 3 |- ( E. x -. x = y -> ( A. x E. x z = y -> E. x A. x z = y ) )
8		hbe1a 2022	. . 3 |- ( E. x A. x z = y -> A. x z = y )
9	2, 7, 8	syl56 36	. 2 |- ( E. x -. x = y -> ( E. x z = y -> A. x z = y ) )
10	1, 9	sylbir 225	1 |- ( -. A. x x = y -> ( E. x z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

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-13 2246

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

This theorem is referenced by: (None)