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Theorem rexab2 3708
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2108. (Revised by GG, 1-Dec-2023.)
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 3069 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓))
2 nfsab1 2720 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1912 . . . 4 𝑦𝜓
42, 3nfan 1897 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} ∧ 𝜓)
5 nfv 1912 . . 3 𝑥(𝜑𝜒)
6 eleq1ab 2714 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2716 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7bitrdi 287 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvexv1 2343 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑𝜒))
121, 11bitri 275 1 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1776  wcel 2106  {cab 2712  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-rex 3069
This theorem is referenced by:  rexrab2  3709  tmdgsum2  24120  clrellem  43612  brtrclfv2  43717
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