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Theorem rexab2 3616
 Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2113. (Revised by Gino Giotto, 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 3076 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓))
2 nfsab1 2744 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1915 . . . 4 𝑦𝜓
42, 3nfan 1900 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} ∧ 𝜓)
5 nfv 1915 . . 3 𝑥(𝜑𝜒)
6 eleq1ab 2737 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2739 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7bitrdi 290 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9anbi12d 633 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvexv1 2351 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑𝜒))
121, 11bitri 278 1 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∃wrex 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-rex 3076 This theorem is referenced by:  rexrab2  3618  tmdgsum2  22810  clrellem  40740  brtrclfv2  40846
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