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Theorem ceqsrexv 3654
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 3068 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
2 an12 645 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
32exbii 1844 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
41, 3bitr4i 278 . 2 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)))
5 eleq1 2826 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 ceqsrexv.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
87ceqsexgv 3653 . . 3 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝐴𝐵𝜓)))
98bianabs 541 . 2 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ 𝜓))
104, 9bitrid 283 1 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068
This theorem is referenced by:  ceqsrexbv  3655  ceqsrex2v  3657  reuxfrd  3756  f1oiso  7370  creur  12257  creui  12258  deg1ldg  26145  ulm2  26442  iscgra1  28832  reuxfrdf  32518  poimirlem24  37630  eqlkr3  39082  diclspsn  41176  rmxdiophlem  43003  expdiophlem1  43009  expdiophlem2  43010
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