Theorem moanim 2260

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

Hypothesis

Ref	Expression
moanim.1	|- F/xph

Assertion

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

Proof of Theorem moanim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 433	. . . . 5 |- (((ph /\ ps) -> x = y) <-> (ph -> (ps -> x = y)))
2	1	albii 1566	. . . 4 |- (A.x((ph /\ ps) -> x = y) <-> A.x(ph -> (ps -> x = y)))
3		moanim.1	. . . . 5 |- F/xph
4	3	19.21 1796	. . . 4 |- (A.x(ph -> (ps -> x = y)) <-> (ph -> A.x(ps -> x = y)))
5	2, 4	bitri 240	. . 3 |- (A.x((ph /\ ps) -> x = y) <-> (ph -> A.x(ps -> x = y)))
6	5	exbii 1582	. 2 |- (E.yA.x((ph /\ ps) -> x = y) <-> E.y(ph -> A.x(ps -> x = y)))
7		nfv 1619	. . 3 |- F/y(ph /\ ps)
8	7	mo2 2233	. 2 |- (E*x(ph /\ ps) <-> E.yA.x((ph /\ ps) -> x = y))
9		nfv 1619	. . . . 5 |- F/yps
10	9	mo2 2233	. . . 4 |- (E*xps <-> E.yA.x(ps -> x = y))
11	10	imbi2i 303	. . 3 |- ((ph -> E*xps) <-> (ph -> E.yA.x(ps -> x = y)))
12		19.37v 1899	. . 3 |- (E.y(ph -> A.x(ps -> x = y)) <-> (ph -> E.yA.x(ps -> x = y)))
13	11, 12	bitr4i 243	. 2 |- ((ph -> E*xps) <-> E.y(ph -> A.x(ps -> x = y)))
14	6, 8, 13	3bitr4i 268	1 |- (E*x(ph /\ ps) <-> (ph -> E*xps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  moanimv  2262  moaneu  2263  moanmo  2264  2eu1  2284