Theorem wl-mo2t 33357

Description: Closed form of mo2 2479. (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 2477	. 2 |- ( E* x ph <-> E. u A. x ( ph -> x = u ) )
2		nfnf1 2031	. . . 4 |- F/ y F/ y ph
3	2	nfal 2153	. . 3 |- F/ y A. x F/ y ph
4		nfa1 2028	. . . 4 |- F/ x A. x F/ y ph
5		sp 2053	. . . . 5 |- ( A. x F/ y ph -> F/ y ph )
6		nfvd 1844	. . . . 5 |- ( A. x F/ y ph -> F/ y x = u )
7	5, 6	nfimd 1823	. . . 4 |- ( A. x F/ y ph -> F/ y ( ph -> x = u ) )
8	4, 7	nfald 2165	. . 3 |- ( A. x F/ y ph -> F/ y A. x ( ph -> x = u ) )
9		equequ2 1953	. . . . . 6 |- ( u = y -> ( x = u <-> x = y ) )
10	9	imbi2d 330	. . . . 5 |- ( u = y -> ( ( ph -> x = u ) <-> ( ph -> x = y ) ) )
11	10	albidv 1849	. . . 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 2278	. 2 |- ( A. x F/ y ph -> ( E. u A. x ( ph -> x = u ) <-> E. y A. x ( ph -> x = y ) ) )
14	1, 13	syl5bb 272	1 |- ( A. x F/ y ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   E*wmo 2471

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  wl-mo3t  33358