Theorem wl-mo2t 31906

Description: Closed form of mo2 2309. (Contributed by Wolf Lammen, 18-Aug-2019.)

Assertion

Ref	Expression
wl-mo2t	|- ( A. x F/ y ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem wl-mo2t

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2307	. 2 |- ( E* x ph <-> E. u A. x ( ph -> x = u ) )
2		nfnf1 1986	. . . 4 |- F/ y F/ y ph
3	2	nfal 2035	. . 3 |- F/ y A. x F/ y ph
4		nfa1 1984	. . . 4 |- F/ x A. x F/ y ph
5		sp 1942	. . . . 5 |- ( A. x F/ y ph -> F/ y ph )
6		nfvd 1766	. . . . 5 |- ( A. x F/ y ph -> F/ y x = u )
7	5, 6	nfimd 2005	. . . 4 |- ( A. x F/ y ph -> F/ y ( ph -> x = u ) )
8	4, 7	nfald 2039	. . 3 |- ( A. x F/ y ph -> F/ y A. x ( ph -> x = u ) )
9		equequ2 1872	. . . . . 6 |- ( u = y -> ( x = u <-> x = y ) )
10	9	imbi2d 322	. . . . 5 |- ( u = y -> ( ( ph -> x = u ) <-> ( ph -> x = y ) ) )
11	10	albidv 1771	. . . 4 |- ( u = y -> ( A. x ( ph -> x = u ) <-> A. x ( ph -> x = y ) ) )
12	11	a1i 11	. . 3 |- ( A. x F/ y ph -> ( u = y -> ( A. x ( ph -> x = u ) <-> A. x ( ph -> x = y ) ) ) )
13	3, 8, 12	cbvexd 2120	. 2 |- ( A. x F/ y ph -> ( E. u A. x ( ph -> x = u ) <-> E. y A. x ( ph -> x = y ) ) )
14	1, 13	syl5bb 265	1 |- ( A. x F/ y ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1446   E.wex 1667   F/wnf 1671   E*wmo 2301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1668  df-nf 1672  df-eu 2304  df-mo 2305

This theorem is referenced by:  wl-mo3t  31907