Theorem wl-mo2t 30185

Description: Closed form of mo2 2229. (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 2225	. 2 |- ( E* x ph <-> E. u A. x ( ph -> x = u ) )
2		nfnf1 1907	. . . 4 |- F/ y F/ y ph
3	2	nfal 1955	. . 3 |- F/ y A. x F/ y ph
4		nfa1 1905	. . . 4 |- F/ x A. x F/ y ph
5		sp 1867	. . . . 5 |- ( A. x F/ y ph -> F/ y ph )
6		nfvd 1716	. . . . 5 |- ( A. x F/ y ph -> F/ y x = u )
7	5, 6	nfimd 1925	. . . 4 |- ( A. x F/ y ph -> F/ y ( ph -> x = u ) )
8	4, 7	nfald 1959	. . 3 |- ( A. x F/ y ph -> F/ y A. x ( ph -> x = u ) )
9		equequ2 1807	. . . . . 6 |- ( u = y -> ( x = u <-> x = y ) )
10	9	imbi2d 314	. . . . 5 |- ( u = y -> ( ( ph -> x = u ) <-> ( ph -> x = y ) ) )
11	10	albidv 1721	. . . 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 2033	. 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 1397   E.wex 1620   F/wnf 1624   E*wmo 2219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-eu 2222  df-mo 2223

This theorem is referenced by:  wl-mo3t  30186