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Theorem rexrab2 3691
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 3145 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21rexeqi 3413 . 2 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43rexab2 3689 . 2 (∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∃𝑦((𝑦𝐴𝜑) ∧ 𝜒))
5 anass 471 . . . 4 (((𝑦𝐴𝜑) ∧ 𝜒) ↔ (𝑦𝐴 ∧ (𝜑𝜒)))
65exbii 1842 . . 3 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
7 df-rex 3142 . . 3 (∃𝑦𝐴 (𝜑𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
86, 7bitr4i 280 . 2 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 299 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1774  wcel 2108  {cab 2797  wrex 3137  {crab 3140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-rex 3142  df-rab 3145
This theorem is referenced by:  frminex  5528  sstotbnd3  35046
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