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Theorem rexab 3336
 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 2902 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒))
2 vex 3176 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 3319 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54anbi1i 727 . . 3 ((𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ (𝜓𝜒))
65exbii 1764 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓𝜒))
71, 6bitri 263 1 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175 This theorem is referenced by:  4sqlem12  15498  nofulllem5  31105  mblfinlem3  32618  mblfinlem4  32619  ismblfin  32620  itg2addnclem  32631  itg2addnc  32634  diophrex  36357
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