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Theorem rexab 3683
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
rexab (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
Distinct variable groups:   𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑥,𝑦)

Proof of Theorem rexab
StepHypRef Expression
1 df-rex 3141 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒))
2 vex 3495 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 3664 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54anbi1i 623 . . 3 ((𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ (𝜓𝜒))
65exbii 1839 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓𝜒))
71, 6bitri 276 1 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wex 1771  wcel 2105  {cab 2796  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494
This theorem is referenced by:  4sqlem12  16280  mblfinlem3  34812  mblfinlem4  34813  ismblfin  34814  itg2addnclem  34824  itg2addnc  34827  diophrex  39250
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