Theorem 2reuswap 1908

Description: A condition allowing swap of uniqueness and existential quantifiers.

Assertion

Ref	Expression
2reuswap	|- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))

Distinct variable group:   x,y,A

Proof of Theorem 2reuswap

Step	Hyp	Ref	Expression
1		df-ral 1625	. . 3 |- (A.x e. A E*y(y e. A /\ ph) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
2		moanimv 1406	. . . 4 |- (E*y(x e. A /\ (y e. A /\ ph)) <-> (x e. A -> E*y(y e. A /\ ph)))
3	2	albii 975	. . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
4	1, 3	bitr4 176	. 2 |- (A.x e. A E*y(y e. A /\ ph) <-> A.xE*y(x e. A /\ (y e. A /\ ph)))
5		2euswap 1422	. . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!xE.y(x e. A /\ (y e. A /\ ph)) -> E!yE.x(x e. A /\ (y e. A /\ ph))))
6		df-reu 1627	. . . 4 |- (E!x e. A E.y e. A ph <-> E!x(x e. A /\ E.y e. A ph))
7		df-rex 1626	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> E.y(y e. A /\ (x e. A /\ ph)))
8		r19.42v 1740	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> (x e. A /\ E.y e. A ph))
9		an12 483	. . . . . . 7 |- ((y e. A /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. A /\ ph)))
10	9	exbii 1027	. . . . . 6 |- (E.y(y e. A /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. A /\ ph)))
11	7, 8, 10	3bitr3 181	. . . . 5 |- ((x e. A /\ E.y e. A ph) <-> E.y(x e. A /\ (y e. A /\ ph)))
12	11	eubii 1364	. . . 4 |- (E!x(x e. A /\ E.y e. A ph) <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
13	6, 12	bitr 173	. . 3 |- (E!x e. A E.y e. A ph <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
14		df-reu 1627	. . . 4 |- (E!y e. A E.x e. A ph <-> E!y(y e. A /\ E.x e. A ph))
15		r19.42v 1740	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> (y e. A /\ E.x e. A ph))
16		df-rex 1626	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
17	15, 16	bitr3 175	. . . . 5 |- ((y e. A /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
18	17	eubii 1364	. . . 4 |- (E!y(y e. A /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
19	14, 18	bitr 173	. . 3 |- (E!y e. A E.x e. A ph <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
20	5, 13, 19	3imtr4g 551	. 2 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))
21	4, 20	sylbi 199	1 |- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950  E.wex 956   e. wcel 1105  E!weu 1357  E*wmo 1358  A.wral 1621  E.wrex 1622  E!wreu 1623

This theorem is referenced by:  reuxfr2 2866

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-ral 1625  df-rex 1626  df-reu 1627

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