Theorem moanim 2093

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 585	. . . . 5 |- ( ( ph /\ ( ps /\ [ y / x ] ps ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) )
2	1	imbi1i 237	. . . 4 |- ( ( ( ph /\ ( ps /\ [ y / x ] ps ) ) -> x = y ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) -> x = y ) )
3		impexp 261	. . . 4 |- ( ( ( ph /\ ( ps /\ [ y / x ] ps ) ) -> x = y ) <-> ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
4		sban 1948	. . . . . . 7 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
5		moanim.1	. . . . . . . . 9 |- F/ x ph
6	5	sbf 1770	. . . . . . . 8 |- ( [ y / x ] ph <-> ph )
7	6	anbi1i 455	. . . . . . 7 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( ph /\ [ y / x ] ps ) )
8	4, 7	bitr2i 184	. . . . . 6 |- ( ( ph /\ [ y / x ] ps ) <-> [ y / x ] ( ph /\ ps ) )
9	8	anbi2i 454	. . . . 5 |- ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) <-> ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) )
10	9	imbi1i 237	. . . 4 |- ( ( ( ( ph /\ ps ) /\ ( ph /\ [ y / x ] ps ) ) -> x = y ) <-> ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
11	2, 3, 10	3bitr3i 209	. . 3 |- ( ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
12	11	2albii 1464	. 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 1576	. . 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 1866	. . . 4 |- ( A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) <-> ( ph -> A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
15	14	albii 1463	. . 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 1519	. . . . 5 |- ( ps -> A. y ps )
17	16	mo3h 2072	. . . 4 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
18	17	imbi2i 225	. . 3 |- ( ( ph -> E* x ps ) <-> ( ph -> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
19	13, 15, 18	3bitr4ri 212	. 2 |- ( ( ph -> E* x ps ) <-> A. x A. y ( ph -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
20		ax-17 1519	. . 3 |- ( ( ph /\ ps ) -> A. y ( ph /\ ps ) )
21	20	mo3h 2072	. 2 |- ( E* x ( ph /\ ps ) <-> A. x A. y ( ( ( ph /\ ps ) /\ [ y / x ] ( ph /\ ps ) ) -> x = y ) )
22	12, 19, 21	3bitr4ri 212	1 |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   F/wnf 1453   [wsb 1755   E*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  moanimv  2094  moaneu  2095  moanmo  2096