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Theorem eueq3 3011
 Description: Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
Hypotheses
Ref Expression
eueq3.1 A V
eueq3.2 B V
eueq3.3 C V
eueq3.4 ¬ (φ ψ)
Assertion
Ref Expression
eueq3 ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))
Distinct variable groups:   φ,x   ψ,x   x,A   x,B   x,C

Proof of Theorem eueq3
StepHypRef Expression
1 eueq3.1 . . . 4 A V
21eueq1 3009 . . 3 ∃!x x = A
3 ibar 490 . . . . . 6 (φ → (x = A ↔ (φ x = A)))
4 pm2.45 386 . . . . . . . . . 10 (¬ (φ ψ) → ¬ φ)
5 eueq3.4 . . . . . . . . . . . 12 ¬ (φ ψ)
65imnani 412 . . . . . . . . . . 11 (φ → ¬ ψ)
76con2i 112 . . . . . . . . . 10 (ψ → ¬ φ)
84, 7jaoi 368 . . . . . . . . 9 ((¬ (φ ψ) ψ) → ¬ φ)
98con2i 112 . . . . . . . 8 (φ → ¬ (¬ (φ ψ) ψ))
104con2i 112 . . . . . . . . . 10 (φ → ¬ ¬ (φ ψ))
1110bianfd 892 . . . . . . . . 9 (φ → (¬ (φ ψ) ↔ (¬ (φ ψ) x = B)))
126bianfd 892 . . . . . . . . 9 (φ → (ψ ↔ (ψ x = C)))
1311, 12orbi12d 690 . . . . . . . 8 (φ → ((¬ (φ ψ) ψ) ↔ ((¬ (φ ψ) x = B) (ψ x = C))))
149, 13mtbid 291 . . . . . . 7 (φ → ¬ ((¬ (φ ψ) x = B) (ψ x = C)))
15 biorf 394 . . . . . . 7 (¬ ((¬ (φ ψ) x = B) (ψ x = C)) → ((φ x = A) ↔ (((¬ (φ ψ) x = B) (ψ x = C)) (φ x = A))))
1614, 15syl 15 . . . . . 6 (φ → ((φ x = A) ↔ (((¬ (φ ψ) x = B) (ψ x = C)) (φ x = A))))
173, 16bitrd 244 . . . . 5 (φ → (x = A ↔ (((¬ (φ ψ) x = B) (ψ x = C)) (φ x = A))))
18 3orrot 940 . . . . . 6 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ((¬ (φ ψ) x = B) (ψ x = C) (φ x = A)))
19 df-3or 935 . . . . . 6 (((¬ (φ ψ) x = B) (ψ x = C) (φ x = A)) ↔ (((¬ (φ ψ) x = B) (ψ x = C)) (φ x = A)))
2018, 19bitri 240 . . . . 5 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ (((¬ (φ ψ) x = B) (ψ x = C)) (φ x = A)))
2117, 20syl6bbr 254 . . . 4 (φ → (x = A ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
2221eubidv 2212 . . 3 (φ → (∃!x x = A∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
232, 22mpbii 202 . 2 (φ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
24 eueq3.3 . . . 4 C V
2524eueq1 3009 . . 3 ∃!x x = C
26 ibar 490 . . . . . 6 (ψ → (x = C ↔ (ψ x = C)))
276adantr 451 . . . . . . . . 9 ((φ x = A) → ¬ ψ)
28 pm2.46 387 . . . . . . . . . 10 (¬ (φ ψ) → ¬ ψ)
2928adantr 451 . . . . . . . . 9 ((¬ (φ ψ) x = B) → ¬ ψ)
3027, 29jaoi 368 . . . . . . . 8 (((φ x = A) (¬ (φ ψ) x = B)) → ¬ ψ)
3130con2i 112 . . . . . . 7 (ψ → ¬ ((φ x = A) (¬ (φ ψ) x = B)))
32 biorf 394 . . . . . . 7 (¬ ((φ x = A) (¬ (φ ψ) x = B)) → ((ψ x = C) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = C))))
3331, 32syl 15 . . . . . 6 (ψ → ((ψ x = C) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = C))))
3426, 33bitrd 244 . . . . 5 (ψ → (x = C ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = C))))
35 df-3or 935 . . . . 5 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = C)))
3634, 35syl6bbr 254 . . . 4 (ψ → (x = C ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
3736eubidv 2212 . . 3 (ψ → (∃!x x = C∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
3825, 37mpbii 202 . 2 (ψ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
39 eueq3.2 . . . 4 B V
4039eueq1 3009 . . 3 ∃!x x = B
41 ibar 490 . . . . . 6 (¬ (φ ψ) → (x = B ↔ (¬ (φ ψ) x = B)))
42 simpl 443 . . . . . . . . 9 ((φ x = A) → φ)
43 simpl 443 . . . . . . . . 9 ((ψ x = C) → ψ)
4442, 43orim12i 502 . . . . . . . 8 (((φ x = A) (ψ x = C)) → (φ ψ))
4544con3i 127 . . . . . . 7 (¬ (φ ψ) → ¬ ((φ x = A) (ψ x = C)))
46 biorf 394 . . . . . . 7 (¬ ((φ x = A) (ψ x = C)) → ((¬ (φ ψ) x = B) ↔ (((φ x = A) (ψ x = C)) (¬ (φ ψ) x = B))))
4745, 46syl 15 . . . . . 6 (¬ (φ ψ) → ((¬ (φ ψ) x = B) ↔ (((φ x = A) (ψ x = C)) (¬ (φ ψ) x = B))))
4841, 47bitrd 244 . . . . 5 (¬ (φ ψ) → (x = B ↔ (((φ x = A) (ψ x = C)) (¬ (φ ψ) x = B))))
49 3orcomb 944 . . . . . 6 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ((φ x = A) (ψ x = C) (¬ (φ ψ) x = B)))
50 df-3or 935 . . . . . 6 (((φ x = A) (ψ x = C) (¬ (φ ψ) x = B)) ↔ (((φ x = A) (ψ x = C)) (¬ (φ ψ) x = B)))
5149, 50bitri 240 . . . . 5 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ (((φ x = A) (ψ x = C)) (¬ (φ ψ) x = B)))
5248, 51syl6bbr 254 . . . 4 (¬ (φ ψ) → (x = B ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
5352eubidv 2212 . . 3 (¬ (φ ψ) → (∃!x x = B∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
5440, 53mpbii 202 . 2 (¬ (φ ψ) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
5523, 38, 54ecase3 907 1 ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  moeq3  3013
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