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Theorem ralrab2 2729
 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 2332 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21raleqi 2526 . 2 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43ralab2 2728 . 2 (∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∀𝑦((𝑦𝐴𝜑) → 𝜒))
5 impexp 254 . . . 4 (((𝑦𝐴𝜑) → 𝜒) ↔ (𝑦𝐴 → (𝜑𝜒)))
65albii 1375 . . 3 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
7 df-ral 2328 . . 3 (∀𝑦𝐴 (𝜑𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
86, 7bitr4i 180 . 2 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 199 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1257   ∈ wcel 1409  {cab 2042  ∀wral 2323  {crab 2327 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332 This theorem is referenced by: (None)
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