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Theorem ceqsex3v 2613
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 387 . . . . . 6 (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
2 3anass 900 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)))
32anbi1i 439 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑))
4 df-3an 898 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑))
54anbi2i 438 . . . . . 6 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
61, 3, 53bitr4i 205 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
762exbii 1513 . . . 4 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
8 19.42vv 1804 . . . 4 (∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
97, 8bitri 177 . . 3 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
109exbii 1512 . 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 1224 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ (𝑦 = 𝐵𝑧 = 𝐶𝜓)))
14132exbidv 1764 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓)))
1511, 14ceqsexv 2610 . . 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 2612 . . 3 (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓) ↔ 𝜃)
2115, 20bitri 177 . 2 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ 𝜃)
2210, 21bitri 177 1 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  ceqsex6v  2615
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