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Theorem ralab2 3598
Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
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 3087 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓))
2 nfsab1 2762 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1873 . . . 4 𝑦𝜓
42, 3nfim 1859 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} → 𝜓)
5 nfv 1873 . . 3 𝑥(𝜑𝜒)
6 eleq1w 2842 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2757 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 279 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9imbi12d 337 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbvalv1 2275 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ ∀𝑦(𝜑𝜒))
121, 11bitri 267 1 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1505  wcel 2048  {cab 2753  wral 3082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-10 2077  ax-11 2091  ax-12 2104
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-clel 2840  df-ral 3087
This theorem is referenced by:  ralrab2  3599  ssintab  4760  efgval  18591  efger  18592  elintabg  39241  elintima  39306
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