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Theorem rexab2 3340
 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
rexab2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
Distinct variable groups:   𝑥,𝑦   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem rexab2
StepHypRef Expression
1 df-rex 2902 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓))
2 nfsab1 2600 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1830 . . . 4 𝑦𝜓
42, 3nfan 1816 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} ∧ 𝜓)
5 nfv 1830 . . 3 𝑥(𝜑𝜒)
6 eleq1 2676 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2598 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 275 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9anbi12d 743 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvex 2260 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑𝜒))
121, 11bitri 263 1 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rex 2902 This theorem is referenced by:  rexrab2  3341  tmdgsum2  21710  clrellem  36948  brtrclfv2  37038
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