Theorem wl-mo2t 31868

Description: Closed form of mo2 2278. (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 2276	. 2 |- ( E* x ph <-> E. u A. x ( ph -> x = u ) )
2		nfnf1 1958	. . . 4 |- F/ y F/ y ph
3	2	nfal 2007	. . 3 |- F/ y A. x F/ y ph
4		nfa1 1956	. . . 4 |- F/ x A. x F/ y ph
5		sp 1914	. . . . 5 |- ( A. x F/ y ph -> F/ y ph )
6		nfvd 1756	. . . . 5 |- ( A. x F/ y ph -> F/ y x = u )
7	5, 6	nfimd 1977	. . . 4 |- ( A. x F/ y ph -> F/ y ( ph -> x = u ) )
8	4, 7	nfald 2011	. . 3 |- ( A. x F/ y ph -> F/ y A. x ( ph -> x = u ) )
9		equequ2 1853	. . . . . 6 |- ( u = y -> ( x = u <-> x = y ) )
10	9	imbi2d 317	. . . . 5 |- ( u = y -> ( ( ph -> x = u ) <-> ( ph -> x = y ) ) )
11	10	albidv 1761	. . . 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 2084	. 2 |- ( A. x F/ y ph -> ( E. u A. x ( ph -> x = u ) <-> E. y A. x ( ph -> x = y ) ) )
14	1, 13	syl5bb 260	1 |- ( A. x F/ y ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661   E*wmo 2270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-eu 2273  df-mo 2274

This theorem is referenced by:  wl-mo3t  31869