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Theorem eueq2dc 2687
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
Hypotheses
Ref Expression
eueq2dc.1 A V
eueq2dc.2 B V
Assertion
Ref Expression
eueq2dc (DECID φ∃!x((φ x = A) φ x = B)))
Distinct variable groups:   φ,x   x,A   x,B

Proof of Theorem eueq2dc
StepHypRef Expression
1 df-dc 731 . 2 (DECID φ ↔ (φ ¬ φ))
2 notnot1 547 . . . . 5 (φ → ¬ ¬ φ)
3 eueq2dc.1 . . . . . . 7 A V
43eueq1 2686 . . . . . 6 ∃!x x = A
5 euanv 1935 . . . . . . 7 (∃!x(φ x = A) ↔ (φ ∃!x x = A))
65biimpri 124 . . . . . 6 ((φ ∃!x x = A) → ∃!x(φ x = A))
74, 6mpan2 403 . . . . 5 (φ∃!x(φ x = A))
8 euorv 1905 . . . . 5 ((¬ ¬ φ ∃!x(φ x = A)) → ∃!xφ (φ x = A)))
92, 7, 8syl2anc 393 . . . 4 (φ∃!xφ (φ x = A)))
10 orcom 634 . . . . . 6 ((¬ φ (φ x = A)) ↔ ((φ x = A) ¬ φ))
112bianfd 841 . . . . . . 7 (φ → (¬ φ ↔ (¬ φ x = B)))
1211orbi2d 691 . . . . . 6 (φ → (((φ x = A) ¬ φ) ↔ ((φ x = A) φ x = B))))
1310, 12syl5bb 181 . . . . 5 (φ → ((¬ φ (φ x = A)) ↔ ((φ x = A) φ x = B))))
1413eubidv 1886 . . . 4 (φ → (∃!xφ (φ x = A)) ↔ ∃!x((φ x = A) φ x = B))))
159, 14mpbid 135 . . 3 (φ∃!x((φ x = A) φ x = B)))
16 eueq2dc.2 . . . . . . 7 B V
1716eueq1 2686 . . . . . 6 ∃!x x = B
18 euanv 1935 . . . . . . 7 (∃!xφ x = B) ↔ (¬ φ ∃!x x = B))
1918biimpri 124 . . . . . 6 ((¬ φ ∃!x x = B) → ∃!xφ x = B))
2017, 19mpan2 403 . . . . 5 φ∃!xφ x = B))
21 euorv 1905 . . . . 5 ((¬ φ ∃!xφ x = B)) → ∃!x(φ φ x = B)))
2220, 21mpdan 400 . . . 4 φ∃!x(φ φ x = B)))
23 id 19 . . . . . . 7 φ → ¬ φ)
2423bianfd 841 . . . . . 6 φ → (φ ↔ (φ x = A)))
2524orbi1d 692 . . . . 5 φ → ((φ φ x = B)) ↔ ((φ x = A) φ x = B))))
2625eubidv 1886 . . . 4 φ → (∃!x(φ φ x = B)) ↔ ∃!x((φ x = A) φ x = B))))
2722, 26mpbid 135 . . 3 φ∃!x((φ x = A) φ x = B)))
2815, 27jaoi 623 . 2 ((φ ¬ φ) → ∃!x((φ x = A) φ x = B)))
291, 28sylbi 114 1 (DECID φ∃!x((φ x = A) φ x = B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616  DECID wdc 730   = wceq 1226   wcel 1370  ∃!weu 1878  Vcvv 2531
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-dc 731  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533
This theorem is referenced by: (None)
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