Theorem wl-mo2t 28536

Description: Closed form of mo2 2272. (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 2267	. 2 |- ( E* x ph <-> E. u A. x ( ph -> x = u ) )
2		nfnf1 1835	. . . 4 |- F/ y F/ y ph
3	2	nfal 1882	. . 3 |- F/ y A. x F/ y ph
4		nfa1 1833	. . . 4 |- F/ x A. x F/ y ph
5		sp 1796	. . . . 5 |- ( A. x F/ y ph -> F/ y ph )
6		nfvd 1675	. . . . 5 |- ( A. x F/ y ph -> F/ y x = u )
7	5, 6	nfimd 1852	. . . 4 |- ( A. x F/ y ph -> F/ y ( ph -> x = u ) )
8	4, 7	nfald 1886	. . 3 |- ( A. x F/ y ph -> F/ y A. x ( ph -> x = u ) )
9		equequ2 1739	. . . . . 6 |- ( u = y -> ( x = u <-> x = y ) )
10	9	imbi2d 316	. . . . 5 |- ( u = y -> ( ( ph -> x = u ) <-> ( ph -> x = y ) ) )
11	10	albidv 1680	. . . 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 1983	. 2 |- ( A. x F/ y ph -> ( E. u A. x ( ph -> x = u ) <-> E. y A. x ( ph -> x = y ) ) )
14	1, 13	syl5bb 257	1 |- ( A. x F/ y ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1368   E.wex 1587   F/wnf 1590   E*wmo 2261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-eu 2264  df-mo 2265

This theorem is referenced by:  wl-mo3t  28537