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Theorem eueq2dc 2779
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((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eueq2dc
StepHypRef Expression
1 df-dc 779 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 notnot 592 . . . . 5 (𝜑 → ¬ ¬ 𝜑)
3 eueq2dc.1 . . . . . . 7 𝐴 ∈ V
43eueq1 2778 . . . . . 6 ∃!𝑥 𝑥 = 𝐴
5 euanv 2002 . . . . . . 7 (∃!𝑥(𝜑𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
65biimpri 131 . . . . . 6 ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑𝑥 = 𝐴))
74, 6mpan2 416 . . . . 5 (𝜑 → ∃!𝑥(𝜑𝑥 = 𝐴))
8 euorv 1972 . . . . 5 ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
92, 7, 8syl2anc 403 . . . 4 (𝜑 → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
10 orcom 680 . . . . . 6 ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑))
112bianfd 892 . . . . . . 7 (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑𝑥 = 𝐵)))
1211orbi2d 737 . . . . . 6 (𝜑 → (((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1310, 12syl5bb 190 . . . . 5 (𝜑 → ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1413eubidv 1953 . . . 4 (𝜑 → (∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
159, 14mpbid 145 . . 3 (𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
16 eueq2dc.2 . . . . . . 7 𝐵 ∈ V
1716eueq1 2778 . . . . . 6 ∃!𝑥 𝑥 = 𝐵
18 euanv 2002 . . . . . . 7 (∃!𝑥𝜑𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
1918biimpri 131 . . . . . 6 ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥𝜑𝑥 = 𝐵))
2017, 19mpan2 416 . . . . 5 𝜑 → ∃!𝑥𝜑𝑥 = 𝐵))
21 euorv 1972 . . . . 5 ((¬ 𝜑 ∧ ∃!𝑥𝜑𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
2220, 21mpdan 412 . . . 4 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
23 id 19 . . . . . . 7 𝜑 → ¬ 𝜑)
2423bianfd 892 . . . . . 6 𝜑 → (𝜑 ↔ (𝜑𝑥 = 𝐴)))
2524orbi1d 738 . . . . 5 𝜑 → ((𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2625eubidv 1953 . . . 4 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2722, 26mpbid 145 . . 3 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
2815, 27jaoi 669 . 2 ((𝜑 ∨ ¬ 𝜑) → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
291, 28sylbi 119 1 (DECID 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 778   = wceq 1287  wcel 1436  ∃!weu 1945  Vcvv 2615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2617
This theorem is referenced by: (None)
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