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Theorem ceqsrex2v 2817
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 398 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
21rexbii 2442 . . . . 5 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 r19.42v 2588 . . . . 5 (∃𝑦𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
42, 3bitri 183 . . . 4 (∃𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
54rexbii 2442 . . 3 (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)))
6 ceqsrex2v.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76anbi2d 459 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
87rexbidv 2438 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝐷 (𝑦 = 𝐵𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
98ceqsrexv 2815 . . 3 (𝐴𝐶 → (∃𝑥𝐶 (𝑥 = 𝐴 ∧ ∃𝑦𝐷 (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
105, 9syl5bb 191 . 2 (𝐴𝐶 → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦𝐷 (𝑦 = 𝐵𝜓)))
11 ceqsrex2v.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1211ceqsrexv 2815 . 2 (𝐵𝐷 → (∃𝑦𝐷 (𝑦 = 𝐵𝜓) ↔ 𝜒))
1310, 12sylan9bb 457 1 ((𝐴𝐶𝐵𝐷) → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2417
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688
This theorem is referenced by: (None)
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