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Theorem eueq3dc 2688
 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 829 . 2 (DECID φ → (DECID ψDECID (φ ψ)))
2 df-dc 731 . . 3 (DECID (φ ψ) ↔ ((φ ψ) ¬ (φ ψ)))
3 eueq3dc.1 . . . . . . 7 A V
43eueq1 2686 . . . . . 6 ∃!x x = A
5 ibar 285 . . . . . . . . 9 (φ → (x = A ↔ (φ x = A)))
6 pm2.45 644 . . . . . . . . . . . . 13 (¬ (φ ψ) → ¬ φ)
7 eueq3dc.4 . . . . . . . . . . . . . . 15 ¬ (φ ψ)
87imnani 612 . . . . . . . . . . . . . 14 (φ → ¬ ψ)
98con2i 545 . . . . . . . . . . . . 13 (ψ → ¬ φ)
106, 9jaoi 623 . . . . . . . . . . . 12 ((¬ (φ ψ) ψ) → ¬ φ)
1110con2i 545 . . . . . . . . . . 11 (φ → ¬ (¬ (φ ψ) ψ))
126con2i 545 . . . . . . . . . . . . 13 (φ → ¬ ¬ (φ ψ))
1312bianfd 841 . . . . . . . . . . . 12 (φ → (¬ (φ ψ) ↔ (¬ (φ ψ) x = B)))
148bianfd 841 . . . . . . . . . . . 12 (φ → (ψ ↔ (ψ x = 𝐶)))
1513, 14orbi12d 694 . . . . . . . . . . 11 (φ → ((¬ (φ ψ) ψ) ↔ ((¬ (φ ψ) x = B) (ψ x = 𝐶))))
1611, 15mtbid 584 . . . . . . . . . 10 (φ → ¬ ((¬ (φ ψ) x = B) (ψ x = 𝐶)))
17 biorf 650 . . . . . . . . . 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 877 . . . . . . . . 9 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ ((¬ (φ ψ) x = B) (ψ x = 𝐶) (φ x = A)))
21 df-3or 872 . . . . . . . . 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 1886 . . . . . 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 2686 . . . . . 6 ∃!x x = 𝐶
28 ibar 285 . . . . . . . . 9 (ψ → (x = 𝐶 ↔ (ψ x = 𝐶)))
298adantr 261 . . . . . . . . . . . 12 ((φ x = A) → ¬ ψ)
30 pm2.46 645 . . . . . . . . . . . . 13 (¬ (φ ψ) → ¬ ψ)
3130adantr 261 . . . . . . . . . . . 12 ((¬ (φ ψ) x = B) → ¬ ψ)
3229, 31jaoi 623 . . . . . . . . . . 11 (((φ x = A) (¬ (φ ψ) x = B)) → ¬ ψ)
3332con2i 545 . . . . . . . . . 10 (ψ → ¬ ((φ x = A) (¬ (φ ψ) x = B)))
34 biorf 650 . . . . . . . . . 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 872 . . . . . . . 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 1886 . . . . . 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 623 . . . 4 ((φ ψ) → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)))
42 eueq3dc.2 . . . . . 6 B V
4342eueq1 2686 . . . . 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 663 . . . . . . . . . 10 (((φ x = A) (ψ x = 𝐶)) → (φ ψ))
4847con3i 549 . . . . . . . . 9 (¬ (φ ψ) → ¬ ((φ x = A) (ψ x = 𝐶)))
49 biorf 650 . . . . . . . . 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 880 . . . . . . . 8 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶)) ↔ ((φ x = A) (ψ x = 𝐶) (¬ (φ ψ) x = B)))
53 df-3or 872 . . . . . . . 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 1886 . . . . 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 623 . . 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 616  DECID wdc 730   ∨ w3o 870   = wceq 1226   ∈ wcel 1370  ∃!weu 1878  Vcvv 2531 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533 This theorem is referenced by:  moeq3dc  2690
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