Theorem bj-mo3OLD 32832

Description: Obsolete proof of mo3 2507 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 2477	. . 3 |- ( E* x ph <-> E. z A. x ( ph -> x = z ) )
2		bj-mo3OLD.nf	. . . . . . . . 9 |- F/ y ph
3		nfv 1843	. . . . . . . . 9 |- F/ y x = z
4	2, 3	nfim 1825	. . . . . . . 8 |- F/ y ( ph -> x = z )
5		nfs1v 2437	. . . . . . . . 9 |- F/ x [ y / x ] ph
6		nfv 1843	. . . . . . . . 9 |- F/ x y = z
7	5, 6	nfim 1825	. . . . . . . 8 |- F/ x ( [ y / x ] ph -> y = z )
8		sbequ2 1882	. . . . . . . . 9 |- ( x = y -> ( [ y / x ] ph -> ph ) )
9		ax7 1943	. . . . . . . . 9 |- ( x = y -> ( x = z -> y = z ) )
10	8, 9	imim12d 81	. . . . . . . 8 |- ( x = y -> ( ( ph -> x = z ) -> ( [ y / x ] ph -> y = z ) ) )
11	4, 7, 10	cbv3 2265	. . . . . . 7 |- ( A. x ( ph -> x = z ) -> A. y ( [ y / x ] ph -> y = z ) )
12	11	ancli 574	. . . . . 6 |- ( A. x ( ph -> x = z ) -> ( A. x ( ph -> x = z ) /\ A. y ( [ y / x ] ph -> y = z ) ) )
13	4, 7	aaan 2170	. . . . . 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 224	. . . . 5 |- ( A. x ( ph -> x = z ) -> A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) )
15		prth 595	. . . . . . 7 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> ( x = z /\ y = z ) ) )
16		equtr2 1954	. . . . . . 7 |- ( ( x = z /\ y = z ) -> x = y )
17	15, 16	syl6 35	. . . . . 6 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
18	17	2alimi 1740	. . . . 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 1858	. . 3 |- ( E. z A. x ( ph -> x = z ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
21	1, 20	sylbi 207	. 2 |- ( E* x ph -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
22		nfa1 2028	. . . . . 6 |- F/ y A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y )
23		pm3.3 460	. . . . . . . . . 10 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ( [ y / x ] ph -> x = y ) ) )
24	23	com3r 87	. . . . . . . . 9 |- ( [ y / x ] ph -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> x = y ) ) )
25	5, 24	alimd 2081	. . . . . . . 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 2055	. . . . . 6 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( [ y / x ] ph -> A. x ( ph -> x = y ) ) )
28	22, 27	eximd 2085	. . . . 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 2425	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
30	2	mo2 2479	. . . . 5 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
31	28, 29, 30	3imtr4g 285	. . . 4 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( E. x ph -> E* x ph ) )
32		moabs 2501	. . . 4 |- ( E* x ph <-> ( E. x ph -> E* x ph ) )
33	31, 32	sylibr 224	. . 3 |- ( A. y A. x ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E* x ph )
34	33	alcoms 2035	. 2 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E* x ph )
35	21, 34	impbii 199	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   [wsb 1880   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-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)