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Theorem ceqsrexv 3642
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 3069 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
2 an12 641 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
32exbii 1848 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
41, 3bitr4i 277 . 2 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)))
5 eleq1 2819 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 ceqsrexv.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 629 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
87ceqsexgv 3641 . . 3 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝐴𝐵𝜓)))
98bianabs 540 . 2 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ 𝜓))
104, 9bitrid 282 1 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rex 3069
This theorem is referenced by:  ceqsrexbv  3643  ceqsrex2v  3645  reuxfrd  3743  f1oiso  7350  creur  12210  creui  12211  deg1ldg  25845  ulm2  26133  iscgra1  28328  reuxfrdf  31998  poimirlem24  36815  eqlkr3  38274  diclspsn  40368  rmxdiophlem  42056  expdiophlem1  42062  expdiophlem2  42063
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