Theorem bj-mo3OLD 31459

Description: Obsolete proof temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-mo3OLD.nf	|- F/ y ph

Assertion

Ref	Expression
bj-mo3OLD	|- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-mo3OLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2308	. . 3 |- ( E* x ph <-> E. z A. x ( ph -> x = z ) )
2		bj-mo3OLD.nf	. . . . . . . . 9 |- F/ y ph
3		nfv 1763	. . . . . . . . 9 |- F/ y x = z
4	2, 3	nfim 2005	. . . . . . . 8 |- F/ y ( ph -> x = z )
5		nfs1v 2268	. . . . . . . . 9 |- F/ x [ y / x ] ph
6		nfv 1763	. . . . . . . . 9 |- F/ x y = z
7	5, 6	nfim 2005	. . . . . . . 8 |- F/ x ( [ y / x ] ph -> y = z )
8		sbequ2 1801	. . . . . . . . 9 |- ( x = y -> ( [ y / x ] ph -> ph ) )
9		ax7 1862	. . . . . . . . 9 |- ( x = y -> ( x = z -> y = z ) )
10	8, 9	imim12d 77	. . . . . . . 8 |- ( x = y -> ( ( ph -> x = z ) -> ( [ y / x ] ph -> y = z ) ) )
11	4, 7, 10	cbv3 2110	. . . . . . 7 |- ( A. x ( ph -> x = z ) -> A. y ( [ y / x ] ph -> y = z ) )
12	11	ancli 554	. . . . . 6 |- ( A. x ( ph -> x = z ) -> ( A. x ( ph -> x = z ) /\ A. y ( [ y / x ] ph -> y = z ) ) )
13	4, 7	aaan 2057	. . . . . 6 |- ( A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) <-> ( A. x ( ph -> x = z ) /\ A. y ( [ y / x ] ph -> y = z ) ) )
14	12, 13	sylibr 216	. . . . 5 |- ( A. x ( ph -> x = z ) -> A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) )
15		prth 575	. . . . . . 7 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> ( x = z /\ y = z ) ) )
16		equtr2 1871	. . . . . . 7 |- ( ( x = z /\ y = z ) -> x = y )
17	15, 16	syl6 34	. . . . . 6 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
18	17	2alimi 1687	. . . . 5 |- ( A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
19	14, 18	syl 17	. . . 4 |- ( A. x ( ph -> x = z ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
20	19	exlimiv 1778	. . 3 |- ( E. z A. x ( ph -> x = z ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
21	1, 20	sylbi 199	. 2 |- ( E* x ph -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
22		nfa1 1981	. . . . . 6 |- F/ y A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y )
23		pm3.3 446	. . . . . . . . . 10 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ( [ y / x ] ph -> x = y ) ) )
24	23	com3r 82	. . . . . . . . 9 |- ( [ y / x ] ph -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> x = y ) ) )
25	5, 24	alimd 1956	. . . . . . . 8 |- ( [ y / x ] ph -> ( A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. x ( ph -> x = y ) ) )
26	25	com12 32	. . . . . . 7 |- ( A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( [ y / x ] ph -> A. x ( ph -> x = y ) ) )
27	26	sps 1945	. . . . . 6 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( [ y / x ] ph -> A. x ( ph -> x = y ) ) )
28	22, 27	eximd 1962	. . . . 5 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( E. y [ y / x ] ph -> E. y A. x ( ph -> x = y ) ) )
29	2	sb8e 2256	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
30	2	mo2 2310	. . . . 5 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
31	28, 29, 30	3imtr4g 274	. . . 4 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( E. x ph -> E* x ph ) )
32		moabs 2332	. . . 4 |- ( E* x ph <-> ( E. x ph -> E* x ph ) )
33	31, 32	sylibr 216	. . 3 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E* x ph )
34	33	alcoms 1923	. 2 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E* x ph )
35	21, 34	impbii 191	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1444   E.wex 1665   F/wnf 1669   [wsb 1799   E*wmo 2302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306

This theorem is referenced by: (None)