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Theorem eueq3dc 2709
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
eueq3dc.1 A V
eueq3dc.2 B V
eueq3dc.3 𝐶 V
eueq3dc.4 ¬ (φ ψ)
Assertion
Ref Expression
eueq3dc (DECID φ → (DECID ψ∃!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 eueq3dc
StepHypRef Expression
1 dcor 842 . 2 (DECID φ → (DECID ψDECID (φ ψ)))
2 df-dc 742 . . 3 (DECID (φ ψ) ↔ ((φ ψ) ¬ (φ ψ)))
3 eueq3dc.1 . . . . . . 7 A V
43eueq1 2707 . . . . . 6 ∃!x x = A
5 ibar 285 . . . . . . . . 9 (φ → (x = A ↔ (φ x = A)))
6 pm2.45 656 . . . . . . . . . . . . 13 (¬ (φ ψ) → ¬ φ)
7 eueq3dc.4 . . . . . . . . . . . . . . 15 ¬ (φ ψ)
87imnani 624 . . . . . . . . . . . . . 14 (φ → ¬ ψ)
98con2i 557 . . . . . . . . . . . . 13 (ψ → ¬ φ)
106, 9jaoi 635 . . . . . . . . . . . 12 ((¬ (φ ψ) ψ) → ¬ φ)
1110con2i 557 . . . . . . . . . . 11 (φ → ¬ (¬ (φ ψ) ψ))
126con2i 557 . . . . . . . . . . . . 13 (φ → ¬ ¬ (φ ψ))
1312bianfd 854 . . . . . . . . . . . 12 (φ → (¬ (φ ψ) ↔ (¬ (φ ψ) x = B)))
148bianfd 854 . . . . . . . . . . . 12 (φ → (ψ ↔ (ψ x = 𝐶)))
1513, 14orbi12d 706 . . . . . . . . . . 11 (φ → ((¬ (φ ψ) ψ) ↔ ((¬ (φ ψ) x = B) (ψ x = 𝐶))))
1611, 15mtbid 596 . . . . . . . . . 10 (φ → ¬ ((¬ (φ ψ) x = B) (ψ x = 𝐶)))
17 biorf 662 . . . . . . . . . 10 (¬ ((¬ (φ ψ) x = B) (ψ x = 𝐶)) → ((φ x = A) ↔ (((¬ (φ ψ) x = B) (ψ x = 𝐶)) (φ x = A))))
1816, 17syl 14 . . . . . . . . 9 (φ → ((φ x = A) ↔ (((¬ (φ ψ) x = B) (ψ x = 𝐶)) (φ x = A))))
195, 18bitrd 177 . . . . . . . 8 (φ → (x = A ↔ (((¬ (φ ψ) x = B) (ψ x = 𝐶)) (φ x = A))))
20 3orrot 890 . . . . . . . . 9 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ ((¬ (φ ψ) x = B) (ψ x = 𝐶) (φ x = A)))
21 df-3or 885 . . . . . . . . 9 (((¬ (φ ψ) x = B) (ψ x = 𝐶) (φ x = A)) ↔ (((¬ (φ ψ) x = B) (ψ x = 𝐶)) (φ x = A)))
2220, 21bitri 173 . . . . . . . 8 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ (((¬ (φ ψ) x = B) (ψ x = 𝐶)) (φ x = A)))
2319, 22syl6bbr 187 . . . . . . 7 (φ → (x = A ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
2423eubidv 1905 . . . . . 6 (φ → (∃!x x = A∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
254, 24mpbii 136 . . . . 5 (φ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
26 eueq3dc.3 . . . . . . 7 𝐶 V
2726eueq1 2707 . . . . . 6 ∃!x x = 𝐶
28 ibar 285 . . . . . . . . 9 (ψ → (x = 𝐶 ↔ (ψ x = 𝐶)))
298adantr 261 . . . . . . . . . . . 12 ((φ x = A) → ¬ ψ)
30 pm2.46 657 . . . . . . . . . . . . 13 (¬ (φ ψ) → ¬ ψ)
3130adantr 261 . . . . . . . . . . . 12 ((¬ (φ ψ) x = B) → ¬ ψ)
3229, 31jaoi 635 . . . . . . . . . . 11 (((φ x = A) (¬ (φ ψ) x = B)) → ¬ ψ)
3332con2i 557 . . . . . . . . . 10 (ψ → ¬ ((φ x = A) (¬ (φ ψ) x = B)))
34 biorf 662 . . . . . . . . . 10 (¬ ((φ x = A) (¬ (φ ψ) x = B)) → ((ψ x = 𝐶) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = 𝐶))))
3533, 34syl 14 . . . . . . . . 9 (ψ → ((ψ x = 𝐶) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = 𝐶))))
3628, 35bitrd 177 . . . . . . . 8 (ψ → (x = 𝐶 ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = 𝐶))))
37 df-3or 885 . . . . . . . 8 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ (((φ x = A) (¬ (φ ψ) x = B)) (ψ x = 𝐶)))
3836, 37syl6bbr 187 . . . . . . 7 (ψ → (x = 𝐶 ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
3938eubidv 1905 . . . . . 6 (ψ → (∃!x x = 𝐶∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
4027, 39mpbii 136 . . . . 5 (ψ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
4125, 40jaoi 635 . . . 4 ((φ ψ) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
42 eueq3dc.2 . . . . . 6 B V
4342eueq1 2707 . . . . 5 ∃!x x = B
44 ibar 285 . . . . . . . 8 (¬ (φ ψ) → (x = B ↔ (¬ (φ ψ) x = B)))
45 simpl 102 . . . . . . . . . . 11 ((φ x = A) → φ)
46 simpl 102 . . . . . . . . . . 11 ((ψ x = 𝐶) → ψ)
4745, 46orim12i 675 . . . . . . . . . 10 (((φ x = A) (ψ x = 𝐶)) → (φ ψ))
4847con3i 561 . . . . . . . . 9 (¬ (φ ψ) → ¬ ((φ x = A) (ψ x = 𝐶)))
49 biorf 662 . . . . . . . . 9 (¬ ((φ x = A) (ψ x = 𝐶)) → ((¬ (φ ψ) x = B) ↔ (((φ x = A) (ψ x = 𝐶)) (¬ (φ ψ) x = B))))
5048, 49syl 14 . . . . . . . 8 (¬ (φ ψ) → ((¬ (φ ψ) x = B) ↔ (((φ x = A) (ψ x = 𝐶)) (¬ (φ ψ) x = B))))
5144, 50bitrd 177 . . . . . . 7 (¬ (φ ψ) → (x = B ↔ (((φ x = A) (ψ x = 𝐶)) (¬ (φ ψ) x = B))))
52 3orcomb 893 . . . . . . . 8 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ ((φ x = A) (ψ x = 𝐶) (¬ (φ ψ) x = B)))
53 df-3or 885 . . . . . . . 8 (((φ x = A) (ψ x = 𝐶) (¬ (φ ψ) x = B)) ↔ (((φ x = A) (ψ x = 𝐶)) (¬ (φ ψ) x = B)))
5452, 53bitri 173 . . . . . . 7 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ (((φ x = A) (ψ x = 𝐶)) (¬ (φ ψ) x = B)))
5551, 54syl6bbr 187 . . . . . 6 (¬ (φ ψ) → (x = B ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
5655eubidv 1905 . . . . 5 (¬ (φ ψ) → (∃!x x = B∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
5743, 56mpbii 136 . . . 4 (¬ (φ ψ) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
5841, 57jaoi 635 . . 3 (((φ ψ) ¬ (φ ψ)) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
592, 58sylbi 114 . 2 (DECID (φ ψ) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
601, 59syl6 29 1 (DECID φ → (DECID ψ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741   w3o 883   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  moeq3dc  2711
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