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Theorem ceqsrexv 2882
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 2474 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
2 an12 561 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
32exbii 1616 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
41, 3bitr4i 187 . 2 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)))
5 eleq1 2252 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 ceqsrexv.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 473 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
87ceqsexgv 2881 . . 3 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝐴𝐵𝜓)))
98bianabs 611 . 2 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝐵𝜑)) ↔ 𝜓))
104, 9bitrid 192 1 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754
This theorem is referenced by:  ceqsrexbv  2883  ceqsrex2v  2884  f1oiso  5848  creur  8946  creui  8947
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