Theorem ax13lem2 2296

Description: Lemma for nfeqf2 2297. This lemma is equivalent to ax13v 2247 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13lem2	|- ( -. x = y -> ( E. x z = y -> z = y ) )

Distinct variable group:   x, z

Proof of Theorem ax13lem2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax13lem1 2248	. . . 4 |- ( -. x = y -> ( w = y -> A. x w = y ) )
2		equeucl 1951	. . . . . 6 |- ( z = y -> ( w = y -> z = w ) )
3	2	eximi 1762	. . . . 5 |- ( E. x z = y -> E. x ( w = y -> z = w ) )
4		19.36v 1904	. . . . 5 |- ( E. x ( w = y -> z = w ) <-> ( A. x w = y -> z = w ) )
5	3, 4	sylib 208	. . . 4 |- ( E. x z = y -> ( A. x w = y -> z = w ) )
6	1, 5	syl9 77	. . 3 |- ( -. x = y -> ( E. x z = y -> ( w = y -> z = w ) ) )
7	6	alrimdv 1857	. 2 |- ( -. x = y -> ( E. x z = y -> A. w ( w = y -> z = w ) ) )
8		equequ2 1953	. . 3 |- ( w = y -> ( z = w <-> z = y ) )
9	8	equsalvw 1931	. 2 |- ( A. w ( w = y -> z = w ) <-> z = y )
10	7, 9	syl6ib 241	1 |- ( -. x = y -> ( E. x z = y -> 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-13 2246

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

This theorem is referenced by:  nfeqf2  2297  wl-speqv  33308  wl-19.2reqv  33310  wl-dveeq12  33311