Theorem moanim 2015

Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Hypothesis

Ref	Expression
moanim.1	|- F/ x ph

Assertion

Ref	Expression
moanim	|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Proof of Theorem moanim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anandi 554	. . . . 5 |- ( ( ph /\ ( ps /\ [ y / x ] ps ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) )
2	1	imbi1i 236	. . . 4 |- ( ( ( ph /\ ( ps /\ [ y / x ] ps ) ) -> x = y ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) -> x = y ) )
3		impexp 259	. . . 4 |- ( ( ( ph /\ ( ps /\ [ y / x ] ps ) ) -> x = y ) <-> ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
4		sban 1870	. . . . . . 7 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
5		moanim.1	. . . . . . . . 9 |- F/ x ph
6	5	sbf 1700	. . . . . . . 8 |- ( [ y / x ] ph <-> ph )
7	6	anbi1i 445	. . . . . . 7 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( ph /\ [ y / x ] ps ) )
8	4, 7	bitr2i 183	. . . . . 6 |- ( ( ph /\ [ y / x ] ps ) <-> [ y / x ] ( ph /\ ps ) )
9	8	anbi2i 444	. . . . 5 |- ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) <-> ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) )
10	9	imbi1i 236	. . . 4 |- ( ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) -> x = y ) <-> ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
11	2, 3, 10	3bitr3i 208	. . 3 |- ( ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
12	11	2albii 1400	. 2 |- ( A. x A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> A. x A. y ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
13	5	19.21 1515	. . 3 |- ( A. x ( ph -> A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> ( ph -> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
14		19.21v 1794	. . . 4 |- ( A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> ( ph -> A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
15	14	albii 1399	. . 3 |- ( A. x A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> A. x ( ph -> A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
16		ax-17 1459	. . . . 5 |- ( ps -> A. y ps )
17	16	mo3h 1994	. . . 4 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
18	17	imbi2i 224	. . 3 |- ( ( ph -> E* x ps ) <-> ( ph -> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
19	13, 15, 18	3bitr4ri 211	. 2 |- ( ( ph -> E* x ps ) <-> A. x A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
20		ax-17 1459	. . 3 |- ( ( ph /\ ps ) -> A. y ( ph /\ ps ) )
21	20	mo3h 1994	. 2 |- ( E* x ( ph /\ ps ) <-> A. x A. y ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
22	12, 19, 21	3bitr4ri 211	1 |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   F/wnf 1389   [wsb 1685   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  moanimv  2016  moaneu  2017  moanmo  2018