Theorem moanim 1429

Description: Introduction of a conjunct into "at most one" quantifier.

Hypothesis

Ref	Expression
moanim.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem moanim

Step	Hyp	Ref	Expression
1		impexp 347	. . . . 5 |- (((ph /\ ps) -> x = y) <-> (ph -> (ps -> x = y)))
2	1	albii 1001	. . . 4 |- (A.x((ph /\ ps) -> x = y) <-> A.x(ph -> (ps -> x = y)))
3		moanim.1	. . . . 5 |- (ph -> A.xph)
4	3	19.21 1058	. . . 4 |- (A.x(ph -> (ps -> x = y)) <-> (ph -> A.x(ps -> x = y)))
5	2, 4	bitr 173	. . 3 |- (A.x((ph /\ ps) -> x = y) <-> (ph -> A.x(ps -> x = y)))
6	5	exbii 1053	. 2 |- (E.yA.x((ph /\ ps) -> x = y) <-> E.y(ph -> A.x(ps -> x = y)))
7		ax-17 973	. . 3 |- ((ph /\ ps) -> A.y(ph /\ ps))
8	7	mo2 1402	. 2 |- (E*x(ph /\ ps) <-> E.yA.x((ph /\ ps) -> x = y))
9		ax-17 973	. . . . 5 |- (ps -> A.yps)
10	9	mo2 1402	. . . 4 |- (E*xps <-> E.yA.x(ps -> x = y))
11	10	imbi2i 185	. . 3 |- ((ph -> E*xps) <-> (ph -> E.yA.x(ps -> x = y)))
12		19.37v 1305	. . 3 |- (E.y(ph -> A.x(ps -> x = y)) <-> (ph -> E.yA.x(ps -> x = y)))
13	11, 12	bitr4 176	. 2 |- ((ph -> E*xps) <-> E.y(ph -> A.x(ps -> x = y)))
14	6, 8, 13	3bitr4 183	1 |- (E*x(ph /\ ps) <-> (ph -> E*xps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956  E.wex 982  E*wmo 1383

This theorem is referenced by:  euan 1430  moanimv 1431  moaneu 1432  moanmo 1433  2eu1 1452

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385

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