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Theorem ceqsrexv 2735
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ceqsrexv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsrexv (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsrexv
StepHypRef Expression
1 df-rex 2359 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
2 an12 526 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
32exbii 1537 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
41, 3bitr4i 185 . 2 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)))
5 eleq1 2145 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 ceqsrexv.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 457 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
87ceqsexgv 2734 . . 3 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝐴𝐵𝜓)))
98bianabs 576 . 2 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ 𝜓))
104, 9syl5bb 190 1 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wrex 2354
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614
This theorem is referenced by:  ceqsrexbv  2736  ceqsrex2v  2737  f1oiso  5544  creur  8313  creui  8314
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