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Theorem ceqsrex2v 2747
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
Hypotheses
Ref Expression
ceqsrex2v.1 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsrex2v.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
ceqsrex2v ((𝐴𝐶𝐵𝐷) → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐶(𝑦)

Proof of Theorem ceqsrex2v
StepHypRef Expression
1 anass 393 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
21rexbii 2385 . . . . 5 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 r19.42v 2524 . . . . 5 (∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
42, 3bitri 182 . . . 4 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
54rexbii 2385 . . 3 (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
6 ceqsrex2v.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76anbi2d 452 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
87rexbidv 2381 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝐷 (𝑦 = 𝐵𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
98ceqsrexv 2745 . . 3 (𝐴𝐶 → (∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
105, 9syl5bb 190 . 2 (𝐴𝐶 → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
11 ceqsrex2v.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1211ceqsrexv 2745 . 2 (𝐵𝐷 → (∃𝑦𝐷 (𝑦 = 𝐵𝜓) ↔ 𝜒))
1310, 12sylan9bb 450 1 ((𝐴𝐶𝐵𝐷) → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621
This theorem is referenced by: (None)
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