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Theorem rexrab2 2820
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 2399 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21rexeqi 2605 . 2 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43rexab2 2819 . 2 (∃𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∃𝑦((𝑦𝐴𝜑) ∧ 𝜒))
5 anass 396 . . . 4 (((𝑦𝐴𝜑) ∧ 𝜒) ↔ (𝑦𝐴 ∧ (𝜑𝜒)))
65exbii 1567 . . 3 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
7 df-rex 2396 . . 3 (∃𝑦𝐴 (𝜑𝜒) ↔ ∃𝑦(𝑦𝐴 ∧ (𝜑𝜒)))
86, 7bitr4i 186 . 2 (∃𝑦((𝑦𝐴𝜑) ∧ 𝜒) ↔ ∃𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 205 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1451  wcel 1463  {cab 2101  wrex 2391  {crab 2394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-rab 2399
This theorem is referenced by: (None)
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