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Theorem ceqsex3v 2729
Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
ceqsex3v.1 𝐴 ∈ V
ceqsex3v.2 𝐵 ∈ V
ceqsex3v.3 𝐶 ∈ V
ceqsex3v.4 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsex3v.5 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex3v.6 (𝑧 = 𝐶 → (𝜒𝜃))
Assertion
Ref Expression
ceqsex3v (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)

Proof of Theorem ceqsex3v
StepHypRef Expression
1 anass 399 . . . . . 6 (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
2 3anass 967 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)))
32anbi1i 454 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑))
4 df-3an 965 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑))
54anbi2i 453 . . . . . 6 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
61, 3, 53bitr4i 211 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
762exbii 1586 . . . 4 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
8 19.42vv 1884 . . . 4 (∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
97, 8bitri 183 . . 3 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
109exbii 1585 . 2 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
11 ceqsex3v.1 . . . 4 𝐴 ∈ V
12 ceqsex3v.4 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
13123anbi3d 1297 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ (𝑦 = 𝐵𝑧 = 𝐶𝜓)))
14132exbidv 1841 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓)))
1511, 14ceqsexv 2726 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓))
16 ceqsex3v.2 . . . 4 𝐵 ∈ V
17 ceqsex3v.3 . . . 4 𝐶 ∈ V
18 ceqsex3v.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
19 ceqsex3v.6 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
2016, 17, 18, 19ceqsex2v 2728 . . 3 (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓) ↔ 𝜃)
2115, 20bitri 183 . 2 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ 𝜃)
2210, 21bitri 183 1 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1332  wex 1469  wcel 1481  Vcvv 2687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2689
This theorem is referenced by:  ceqsex6v  2731
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