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Theorem ralrab2 3358
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 2916 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21raleqi 3134 . 2 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43ralab2 3357 . 2 (∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∀𝑦((𝑦𝐴𝜑) → 𝜒))
5 impexp 462 . . . 4 (((𝑦𝐴𝜑) → 𝜒) ↔ (𝑦𝐴 → (𝜑𝜒)))
65albii 1744 . . 3 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
7 df-ral 2912 . . 3 (∀𝑦𝐴 (𝜑𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
86, 7bitr4i 267 . 2 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 286 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  {cab 2607  wral 2907  {crab 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916
This theorem is referenced by:  efgsf  18074  ghmcnp  21841  nmogelb  22443  pntlem3  25215  sstotbnd2  33240
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