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Theorem rexrab 2704
 Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
rexrab (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rexrab
StepHypRef Expression
1 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
21elrab 2698 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32anbi1i 431 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ ((𝑥𝐴𝜓) ∧ 𝜒))
4 anass 381 . . 3 (((𝑥𝐴𝜓) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
53, 4bitri 173 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
65rexbii2 2335 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∃wrex 2307  {crab 2310 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559 This theorem is referenced by: (None)
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