Theorem euanv 1053

Description: Introduction of a conjunct into uniqueness quantifier.

Assertion

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

Distinct variable group(s):   ph,x

Proof of Theorem euanv

Step	Hyp	Ref	Expression
1		19.42v 966	. . . 4 |- (E.x(ph /\ ps) <-> (ph /\ E.xps))
2		moanimv 1052	. . . . . 6 |- (E*x(ph /\ ps) <-> (ph -> E*xps))
3	2	anbi2i 367	. . . . 5 |- ((ph /\ E*x(ph /\ ps)) <-> (ph /\ (ph -> E*xps)))
4		abai 366	. . . . 5 |- ((ph /\ E*xps) <-> (ph /\ (ph -> E*xps)))
5	3, 4	bitr4 154	. . . 4 |- ((ph /\ E*x(ph /\ ps)) <-> (ph /\ E*xps))
6	1, 5	anbi12i 369	. . 3 |- ((E.x(ph /\ ps) /\ (ph /\ E*x(ph /\ ps))) <-> ((ph /\ E.xps) /\ (ph /\ E*xps)))
7		anass 336	. . 3 |- (((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)) <-> (E.x(ph /\ ps) /\ (ph /\ E*x(ph /\ ps))))
8		an4 388	. . 3 |- (((ph /\ ph) /\ (E.xps /\ E*xps)) <-> ((ph /\ E.xps) /\ (ph /\ E*xps)))
9	6, 7, 8	3bitr4 158	. 2 |- (((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)) <-> ((ph /\ ph) /\ (E.xps /\ E*xps)))
10		eu5 1035	. . 3 |- (E!x(ph /\ ps) <-> (E.x(ph /\ ps) /\ E*x(ph /\ ps)))
11		anabs1 374	. . . . . 6 |- (((ph /\ ps) /\ ph) <-> (ph /\ ps))
12	11	biex 733	. . . . 5 |- (E.x((ph /\ ps) /\ ph) <-> E.x(ph /\ ps))
13		19.41v 963	. . . . 5 |- (E.x((ph /\ ps) /\ ph) <-> (E.x(ph /\ ps) /\ ph))
14	12, 13	bitr3 153	. . . 4 |- (E.x(ph /\ ps) <-> (E.x(ph /\ ps) /\ ph))
15	14	anbi1i 368	. . 3 |- ((E.x(ph /\ ps) /\ E*x(ph /\ ps)) <-> ((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)))
16	10, 15	bitr 151	. 2 |- (E!x(ph /\ ps) <-> ((E.x(ph /\ ps) /\ ph) /\ E*x(ph /\ ps)))
17		anidm 331	. . . 4 |- ((ph /\ ph) <-> ph)
18	17	bicomi 150	. . 3 |- (ph <-> (ph /\ ph))
19		eu5 1035	. . 3 |- (E!xps <-> (E.xps /\ E*xps))
20	18, 19	anbi12i 369	. 2 |- ((ph /\ E!xps) <-> ((ph /\ ph) /\ (E.xps /\ E*xps)))
21	9, 16, 20	3bitr4 158	1 |- (E!x(ph /\ ps) <-> (ph /\ E!xps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678  E!weu 1007  E*wmo 1008

This theorem is referenced by:  eueq2 1429  eueq3 1430  fnopabg 2745  fvopab2 2878  fsn 2895  aceq5lem5 3562

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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