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Theorem eueq2dc 2852
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 820 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 notnot 618 . . . . 5 (𝜑 → ¬ ¬ 𝜑)
3 eueq2dc.1 . . . . . . 7 𝐴 ∈ V
43eueq1 2851 . . . . . 6 ∃!𝑥 𝑥 = 𝐴
5 euanv 2054 . . . . . . 7 (∃!𝑥(𝜑𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
65biimpri 132 . . . . . 6 ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑𝑥 = 𝐴))
74, 6mpan2 421 . . . . 5 (𝜑 → ∃!𝑥(𝜑𝑥 = 𝐴))
8 euorv 2024 . . . . 5 ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
92, 7, 8syl2anc 408 . . . 4 (𝜑 → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
10 orcom 717 . . . . . 6 ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑))
112bianfd 932 . . . . . . 7 (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑𝑥 = 𝐵)))
1211orbi2d 779 . . . . . 6 (𝜑 → (((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1310, 12syl5bb 191 . . . . 5 (𝜑 → ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1413eubidv 2005 . . . 4 (𝜑 → (∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
159, 14mpbid 146 . . 3 (𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
16 eueq2dc.2 . . . . . . 7 𝐵 ∈ V
1716eueq1 2851 . . . . . 6 ∃!𝑥 𝑥 = 𝐵
18 euanv 2054 . . . . . . 7 (∃!𝑥𝜑𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
1918biimpri 132 . . . . . 6 ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥𝜑𝑥 = 𝐵))
2017, 19mpan2 421 . . . . 5 𝜑 → ∃!𝑥𝜑𝑥 = 𝐵))
21 euorv 2024 . . . . 5 ((¬ 𝜑 ∧ ∃!𝑥𝜑𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
2220, 21mpdan 417 . . . 4 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
23 id 19 . . . . . . 7 𝜑 → ¬ 𝜑)
2423bianfd 932 . . . . . 6 𝜑 → (𝜑 ↔ (𝜑𝑥 = 𝐴)))
2524orbi1d 780 . . . . 5 𝜑 → ((𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2625eubidv 2005 . . . 4 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2722, 26mpbid 146 . . 3 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
2815, 27jaoi 705 . 2 ((𝜑 ∨ ¬ 𝜑) → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
291, 28sylbi 120 1 (DECID 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  ∃!weu 1997  Vcvv 2681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by: (None)
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