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Theorem ceqsrexbv 2790
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
Hypothesis
Ref Expression
ceqsrexv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsrexbv (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsrexbv
StepHypRef Expression
1 r19.42v 2565 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)))
2 eleq1 2180 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 274 . . . . . 6 ((𝑥 = 𝐴𝜑) → (𝑥𝐵𝐴𝐵))
43pm5.32ri 450 . . . . 5 ((𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)))
54bicomi 131 . . . 4 ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
65baib 889 . . 3 (𝑥𝐵 → ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥 = 𝐴𝜑)))
76rexbiia 2427 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ ∃𝑥𝐵 (𝑥 = 𝐴𝜑))
8 ceqsrexv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
98ceqsrexv 2789 . . 3 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
109pm5.32i 449 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵𝜓))
111, 7, 103bitr3i 209 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wrex 2394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by:  frecsuclem  6271
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