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Theorem ralrab2 3704
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 3437 . . 3 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21raleqi 3324 . 2 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓)
3 ralab2.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
43ralab2 3703 . 2 (∀𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)}𝜓 ↔ ∀𝑦((𝑦𝐴𝜑) → 𝜒))
5 impexp 450 . . . 4 (((𝑦𝐴𝜑) → 𝜒) ↔ (𝑦𝐴 → (𝜑𝜒)))
65albii 1819 . . 3 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
7 df-ral 3062 . . 3 (∀𝑦𝐴 (𝜑𝜒) ↔ ∀𝑦(𝑦𝐴 → (𝜑𝜒)))
86, 7bitr4i 278 . 2 (∀𝑦((𝑦𝐴𝜑) → 𝜒) ↔ ∀𝑦𝐴 (𝜑𝜒))
92, 4, 83bitri 297 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  {cab 2714  wral 3061  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-ral 3062  df-rex 3071  df-rab 3437
This theorem is referenced by:  efgsf  19747  ghmcnp  24123  nmogelb  24737  pntlem3  27653  sstotbnd2  37781
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