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Theorem rexab 2723
 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 2327 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒))
2 vex 2575 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 2707 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54anbi1i 439 . . 3 ((𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ (𝜓𝜒))
65exbii 1510 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓𝜒))
71, 6bitri 177 1 (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∃wex 1395   ∈ wcel 1407  {cab 2040  ∃wrex 2322 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574 This theorem is referenced by:  rexrnmpt2  5641
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