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Theorem 2reuswapdc 2737
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
2reuswapdc (DECID xy(x A (y B φ)) → (x A ∃*y B φ → (∃!x A y B φ∃!y B x A φ)))
Distinct variable groups:   x,y,A   x,B
Allowed substitution hints:   φ(x,y)   B(y)

Proof of Theorem 2reuswapdc
StepHypRef Expression
1 df-rmo 2308 . . 3 (∃*y B φ∃*y(y B φ))
21ralbii 2324 . 2 (x A ∃*y B φx A ∃*y(y B φ))
3 df-ral 2305 . . . 4 (x A ∃*y(y B φ) ↔ x(x A∃*y(y B φ)))
4 moanimv 1972 . . . . 5 (∃*y(x A (y B φ)) ↔ (x A∃*y(y B φ)))
54albii 1356 . . . 4 (x∃*y(x A (y B φ)) ↔ x(x A∃*y(y B φ)))
63, 5bitr4i 176 . . 3 (x A ∃*y(y B φ) ↔ x∃*y(x A (y B φ)))
7 df-reu 2307 . . . . . 6 (∃!x A y B φ∃!x(x A y B φ))
8 r19.42v 2461 . . . . . . . . 9 (y B (x A φ) ↔ (x A y B φ))
9 df-rex 2306 . . . . . . . . 9 (y B (x A φ) ↔ y(y B (x A φ)))
108, 9bitr3i 175 . . . . . . . 8 ((x A y B φ) ↔ y(y B (x A φ)))
11 an12 495 . . . . . . . . 9 ((y B (x A φ)) ↔ (x A (y B φ)))
1211exbii 1493 . . . . . . . 8 (y(y B (x A φ)) ↔ y(x A (y B φ)))
1310, 12bitri 173 . . . . . . 7 ((x A y B φ) ↔ y(x A (y B φ)))
1413eubii 1906 . . . . . 6 (∃!x(x A y B φ) ↔ ∃!xy(x A (y B φ)))
157, 14bitri 173 . . . . 5 (∃!x A y B φ∃!xy(x A (y B φ)))
16 2euswapdc 1988 . . . . 5 (DECID xy(x A (y B φ)) → (x∃*y(x A (y B φ)) → (∃!xy(x A (y B φ)) → ∃!yx(x A (y B φ)))))
1715, 16syl7bi 154 . . . 4 (DECID xy(x A (y B φ)) → (x∃*y(x A (y B φ)) → (∃!x A y B φ∃!yx(x A (y B φ)))))
18 df-reu 2307 . . . . . 6 (∃!y B x A φ∃!y(y B x A φ))
19 r19.42v 2461 . . . . . . . 8 (x A (y B φ) ↔ (y B x A φ))
20 df-rex 2306 . . . . . . . 8 (x A (y B φ) ↔ x(x A (y B φ)))
2119, 20bitr3i 175 . . . . . . 7 ((y B x A φ) ↔ x(x A (y B φ)))
2221eubii 1906 . . . . . 6 (∃!y(y B x A φ) ↔ ∃!yx(x A (y B φ)))
2318, 22bitri 173 . . . . 5 (∃!y B x A φ∃!yx(x A (y B φ)))
2423imbi2i 215 . . . 4 ((∃!x A y B φ∃!y B x A φ) ↔ (∃!x A y B φ∃!yx(x A (y B φ))))
2517, 24syl6ibr 151 . . 3 (DECID xy(x A (y B φ)) → (x∃*y(x A (y B φ)) → (∃!x A y B φ∃!y B x A φ)))
266, 25syl5bi 141 . 2 (DECID xy(x A (y B φ)) → (x A ∃*y(y B φ) → (∃!x A y B φ∃!y B x A φ)))
272, 26syl5bi 141 1 (DECID xy(x A (y B φ)) → (x A ∃*y B φ → (∃!x A y B φ∃!y B x A φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  DECID wdc 741  wal 1240  wex 1378   wcel 1390  ∃!weu 1897  ∃*wmo 1898  wral 2300  wrex 2301  ∃!wreu 2302  ∃*wrmo 2303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308
This theorem is referenced by: (None)
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