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Theorem rexab2 3580
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 3113 . 2 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓))
2 nfsab1 2807 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 2005 . . . 4 𝑦𝜓
42, 3nfan 1990 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} ∧ 𝜓)
5 nfv 2005 . . 3 𝑥(𝜑𝜒)
6 eleq1w 2879 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2805 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 278 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9anbi12d 618 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvex 2447 . 2 (∃𝑥(𝑥 ∈ {𝑦𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑𝜒))
121, 11bitri 266 1 (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wex 1859  wcel 2157  {cab 2803  wrex 3108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-clel 2813  df-rex 3113
This theorem is referenced by:  rexrab2  3581  tmdgsum2  22133  clrellem  38446  brtrclfv2  38536
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