Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eueq2 Structured version   Visualization version   GIF version

Theorem eueq2 3699
 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 144 . . . 4 (𝜑 → ¬ ¬ 𝜑)
2 eueq2.1 . . . . . 6 𝐴 ∈ V
32eueqi 3698 . . . . 5 ∃!𝑥 𝑥 = 𝐴
4 euanv 2703 . . . . . 6 (∃!𝑥(𝜑𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
54biimpri 230 . . . . 5 ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑𝑥 = 𝐴))
63, 5mpan2 689 . . . 4 (𝜑 → ∃!𝑥(𝜑𝑥 = 𝐴))
7 euorv 2690 . . . 4 ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
81, 6, 7syl2anc 586 . . 3 (𝜑 → ∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)))
9 orcom 866 . . . . 5 ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑))
101bianfd 537 . . . . . 6 (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑𝑥 = 𝐵)))
1110orbi2d 912 . . . . 5 (𝜑 → (((𝜑𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
129, 11syl5bb 285 . . . 4 (𝜑 → ((¬ 𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
1312eubidv 2666 . . 3 (𝜑 → (∃!𝑥𝜑 ∨ (𝜑𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
148, 13mpbid 234 . 2 (𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
15 eueq2.2 . . . . . 6 𝐵 ∈ V
1615eueqi 3698 . . . . 5 ∃!𝑥 𝑥 = 𝐵
17 euanv 2703 . . . . . 6 (∃!𝑥𝜑𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
1817biimpri 230 . . . . 5 ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥𝜑𝑥 = 𝐵))
1916, 18mpan2 689 . . . 4 𝜑 → ∃!𝑥𝜑𝑥 = 𝐵))
20 euorv 2690 . . . 4 ((¬ 𝜑 ∧ ∃!𝑥𝜑𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
2119, 20mpdan 685 . . 3 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)))
22 id 22 . . . . . 6 𝜑 → ¬ 𝜑)
2322bianfd 537 . . . . 5 𝜑 → (𝜑 ↔ (𝜑𝑥 = 𝐴)))
2423orbi1d 913 . . . 4 𝜑 → ((𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2524eubidv 2666 . . 3 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))))
2621, 25mpbid 234 . 2 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
2714, 26pm2.61i 184 1 ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108  ∃!weu 2647  Vcvv 3493 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-v 3495 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator