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Theorem rexrab2 3407
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (𝑥 = 𝑦 → (𝜓𝜒))
Assertion
Ref Expression
rexrab2 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem rexrab2
StepHypRef Expression
1 df-rab 2950 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21rexeqi 3173 . 2 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43rexab2 3406 . 2 (∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∃𝑦((𝑦𝐴𝜑) ∧ 𝜒))
5 anass 682 . . . 4 (((𝑦𝐴𝜑) ∧ 𝜒) ↔ (𝑦𝐴 ∧ (𝜑𝜒)))
65exbii 1814 . . 3 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
7 df-rex 2947 . . 3 (∃𝑦𝐴 (𝜑𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
86, 7bitr4i 267 . 2 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 286 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1744  wcel 2030  {cab 2637  wrex 2942  {crab 2945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950
This theorem is referenced by:  frminex  5123  sstotbnd3  33705
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