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Theorem ceqsrex2v 2676
 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 381 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
21rexbii 2331 . . . . 5 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 r19.42v 2467 . . . . 5 (∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
42, 3bitri 173 . . . 4 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
54rexbii 2331 . . 3 (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
6 ceqsrex2v.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76anbi2d 437 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
87rexbidv 2327 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝐷 (𝑦 = 𝐵𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
98ceqsrexv 2674 . . 3 (𝐴𝐶 → (∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
105, 9syl5bb 181 . 2 (𝐴𝐶 → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
11 ceqsrex2v.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1211ceqsrexv 2674 . 2 (𝐵𝐷 → (∃𝑦𝐷 (𝑦 = 𝐵𝜓) ↔ 𝜒))
1310, 12sylan9bb 435 1 ((𝐴𝐶𝐵𝐷) → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559 This theorem is referenced by: (None)
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