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Theorem rexab 2777
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 2365 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒))
2 vex 2622 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 2760 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54anbi1i 446 . . 3 ((𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ (𝜓𝜒))
65exbii 1541 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓𝜒))
71, 6bitri 182 1 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1426  wcel 1438  {cab 2074  wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621
This theorem is referenced by:  rexrnmpt2  5752
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