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Theorem ralab2 3639
Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2114. (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 3114 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓))
2 nfsab1 2788 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1915 . . . 4 𝑦𝜓
42, 3nfim 1897 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} → 𝜓)
5 nfv 1915 . . 3 𝑥(𝜑𝜒)
6 eleq1ab 2781 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2783 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 290 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9imbi12d 348 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvalv1 2353 . 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  wal 1536  wcel 2112  {cab 2779  wral 3109
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 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-ral 3114
This theorem is referenced by:  ralrab2  3641  ssintab  4858  efgval  18839  efger  18840  elintabg  40267  elintima  40347
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