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Theorem ralrab2 2802
Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (𝑥 = 𝑦 → (𝜓𝜒))
Assertion
Ref Expression
ralrab2 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem ralrab2
StepHypRef Expression
1 df-rab 2384 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21raleqi 2588 . 2 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43ralab2 2801 . 2 (∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∀𝑦((𝑦𝐴𝜑) → 𝜒))
5 impexp 261 . . . 4 (((𝑦𝐴𝜑) → 𝜒) ↔ (𝑦𝐴 → (𝜑𝜒)))
65albii 1414 . . 3 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
7 df-ral 2380 . . 3 (∀𝑦𝐴 (𝜑𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
86, 7bitr4i 186 . 2 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 205 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1297  wcel 1448  {cab 2086  wral 2375  {crab 2379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384
This theorem is referenced by: (None)
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