Theorem euan 1428

Description: Introduction of a conjunct into uniqueness quantifier.

Hypothesis

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

Assertion

Ref	Expression
euan	|- (E!x(ph /\ ps) <-> (ph /\ E!xps))

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		moanim.1	. . . . 5 |- (ph -> A.xph)
2	1	19.42 1096	. . . 4 |- (E.x(ph /\ ps) <-> (ph /\ E.xps))
3	1	moanim 1427	. . . . . 6 |- (E*x(ph /\ ps) <-> (ph -> E*xps))
4	3	anbi2i 480	. . . . 5 |- ((ph /\ E*x(ph /\ ps)) <-> (ph /\ (ph -> E*xps)))
5		abai 479	. . . . 5 |- ((ph /\ E*xps) <-> (ph /\ (ph -> E*xps)))
6	4, 5	bitr4 176	. . . 4 |- ((ph /\ E*x(ph /\ ps)) <-> (ph /\ E*xps))
7	2, 6	anbi12i 482	. . 3 |- ((E.x(ph /\ ps) /\ (ph /\ E*x(ph /\ ps))) <-> ((ph /\ E.xps) /\ (ph /\ E*xps)))
8		anass 439	. . 3 |- (((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)) <-> (E.x(ph /\ ps) /\ (ph /\ E*x(ph /\ ps))))
9		an4 506	. . 3 |- (((ph /\ ph) /\ (E.xps /\ E*xps)) <-> ((ph /\ E.xps) /\ (ph /\ E*xps)))
10	7, 8, 9	3bitr4 183	. 2 |- (((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)) <-> ((ph /\ ph) /\ (E.xps /\ E*xps)))
11		eu5 1409	. . 3 |- (E!x(ph /\ ps) <-> (E.x(ph /\ ps) /\ E*x(ph /\ ps)))
12		anabs1 492	. . . . . 6 |- (((ph /\ ps) /\ ph) <-> (ph /\ ps))
13	12	exbii 1051	. . . . 5 |- (E.x((ph /\ ps) /\ ph) <-> E.x(ph /\ ps))
14	1	19.41 1095	. . . . 5 |- (E.x((ph /\ ps) /\ ph) <-> (E.x(ph /\ ps) /\ ph))
15	13, 14	bitr3 175	. . . 4 |- (E.x(ph /\ ps) <-> (E.x(ph /\ ps) /\ ph))
16	15	anbi1i 481	. . 3 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) <-> ((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)))
17	11, 16	bitr 173	. 2 |- (E!x(ph /\ ps) <-> ((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)))
18		pm4.24 433	. . 3 |- (ph <-> (ph /\ ph))
19		eu5 1409	. . 3 |- (E!xps <-> (E.xps /\ E*xps))
20	18, 19	anbi12i 482	. 2 |- ((ph /\ E!xps) <-> ((ph /\ ph) /\ (E.xps /\ E*xps)))
21	10, 17, 20	3bitr4 183	1 |- (E!x(ph /\ ps) <-> (ph /\ E!xps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954  E.wex 980  E!weu 1380  E*wmo 1381

This theorem is referenced by:  euanv 1432  2eu7 1455  2eu8 1456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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