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Theorem rexab2 3406
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 2947 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓))
2 nfsab1 2641 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1883 . . . 4 𝑦𝜓
42, 3nfan 1868 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} ∧ 𝜓)
5 nfv 1883 . . 3 𝑥(𝜑𝜒)
6 eleq1 2718 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2639 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 276 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9anbi12d 747 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvex 2308 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑𝜒))
121, 11bitri 264 1 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1744  wcel 2030  {cab 2637  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-rex 2947
This theorem is referenced by:  rexrab2  3407  tmdgsum2  21947  clrellem  38246  brtrclfv2  38336
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