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Theorem ralab2 3694
Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2109. (Revised by Gino Giotto, 1-Dec-2023.)
Hypothesis
Ref Expression
ralab2.1 (𝑥 = 𝑦 → (𝜓𝜒))
Assertion
Ref Expression
ralab2 (∀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 ralab2
StepHypRef Expression
1 df-ral 3063 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓))
2 nfsab1 2718 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1918 . . . 4 𝑦𝜓
42, 3nfim 1900 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} → 𝜓)
5 nfv 1918 . . 3 𝑥(𝜑𝜒)
6 eleq1ab 2712 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2714 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7bitrdi 287 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9imbi12d 345 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvalv1 2338 . 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 205  wal 1540  wcel 2107  {cab 2710  wral 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-ral 3063
This theorem is referenced by:  ralrab2  3695  elintabg  4962  ssintab  4970  efgval  19585  efger  19586  elintima  42404
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