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Theorem rexrab2 3361
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 2921 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21rexeqi 3137 . 2 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43rexab2 3360 . 2 (∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∃𝑦((𝑦𝐴𝜑) ∧ 𝜒))
5 anass 680 . . . 4 (((𝑦𝐴𝜑) ∧ 𝜒) ↔ (𝑦𝐴 ∧ (𝜑𝜒)))
65exbii 1772 . . 3 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
7 df-rex 2918 . . 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 384  wex 1701  wcel 1992  {cab 2612  wrex 2913  {crab 2916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-rab 2921
This theorem is referenced by:  frminex  5059  sstotbnd3  33193
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