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Theorem eueq2 3367
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
Hypotheses
Ref Expression
eueq2.1 𝐴 ∈ V
eueq2.2 𝐵 ∈ V
Assertion
Ref Expression
eueq2 ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eueq2
StepHypRef Expression
1 notnot 136 . . . 4 (𝜑 → ¬ ¬ 𝜑)
2 eueq2.1 . . . . . 6 𝐴 ∈ V
32eueq1 3366 . . . . 5 ∃!𝑥 𝑥 = 𝐴
4 euanv 2533 . . . . . 6 (∃!𝑥(𝜑𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
54biimpri 218 . . . . 5 ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑𝑥 = 𝐴))
63, 5mpan2 706 . . . 4 (𝜑 → ∃!𝑥(𝜑𝑥 = 𝐴))
7 euorv 2512 . . . 4 ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
81, 6, 7syl2anc 692 . . 3 (𝜑 → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
9 orcom 402 . . . . 5 ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑))
101bianfd 966 . . . . . 6 (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑𝑥 = 𝐵)))
1110orbi2d 737 . . . . 5 (𝜑 → (((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
129, 11syl5bb 272 . . . 4 (𝜑 → ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1312eubidv 2489 . . 3 (𝜑 → (∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
148, 13mpbid 222 . 2 (𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
15 eueq2.2 . . . . . 6 𝐵 ∈ V
1615eueq1 3366 . . . . 5 ∃!𝑥 𝑥 = 𝐵
17 euanv 2533 . . . . . 6 (∃!𝑥𝜑𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
1817biimpri 218 . . . . 5 ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥𝜑𝑥 = 𝐵))
1916, 18mpan2 706 . . . 4 𝜑 → ∃!𝑥𝜑𝑥 = 𝐵))
20 euorv 2512 . . . 4 ((¬ 𝜑 ∧ ∃!𝑥𝜑𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
2119, 20mpdan 701 . . 3 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
22 id 22 . . . . . 6 𝜑 → ¬ 𝜑)
2322bianfd 966 . . . . 5 𝜑 → (𝜑 ↔ (𝜑𝑥 = 𝐴)))
2423orbi1d 738 . . . 4 𝜑 → ((𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2524eubidv 2489 . . 3 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2621, 25mpbid 222 . 2 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
2714, 26pm2.61i 176 1 ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383  wa 384   = wceq 1480  wcel 1987  ∃!weu 2469  Vcvv 3190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192
This theorem is referenced by: (None)
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