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Theorem rexab 3521
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 3067 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒))
2 vex 3354 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 3501 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54anbi1i 602 . . 3 ((𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ (𝜓𝜒))
65exbii 1924 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓𝜒))
71, 6bitri 264 1 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wex 1852  wcel 2145  {cab 2757  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353
This theorem is referenced by:  4sqlem12  15866  mblfinlem3  33777  mblfinlem4  33778  ismblfin  33779  itg2addnclem  33789  itg2addnc  33792  diophrex  37861
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