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Theorem ralab2 2821
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 2398 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓))
2 nfsab1 2107 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1493 . . . 4 𝑦𝜓
42, 3nfim 1536 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} → 𝜓)
5 nfv 1493 . . 3 𝑥(𝜑𝜒)
6 eleq1 2180 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2105 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 195 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9imbi12d 233 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbval 1712 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ ∀𝑦(𝜑𝜒))
121, 11bitri 183 1 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wcel 1465  {cab 2103  wral 2393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398
This theorem is referenced by:  ralrab2  2822  ssintab  3758
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