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Theorem rabeq0w 4393
Description: Condition for a restricted class abstraction to be empty. Version of rabeq0 4394 using implicit substitution, which does not require ax-10 2139, ax-11 2155, ax-12 2175, but requires ax-8 2108. (Contributed by GG, 30-Sep-2024.)
Hypothesis
Ref Expression
rabeq0w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
rabeq0w ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦𝐴 ¬ 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rabeq0w
StepHypRef Expression
1 eleq1w 2822 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 rabeq0w.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43ab0w 4385 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} = ∅ ↔ ∀𝑦 ¬ (𝑦𝐴𝜓))
5 df-rab 3434 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eqeq1i 2740 . 2 ({𝑥𝐴𝜑} = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = ∅)
7 raln 3067 . 2 (∀𝑦𝐴 ¬ 𝜓 ↔ ∀𝑦 ¬ (𝑦𝐴𝜓))
84, 6, 73bitr4i 303 1 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦𝐴 ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  {crab 3433  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-dif 3966  df-nul 4340
This theorem is referenced by:  dffr2  5650  frc  5652  frirr  5665
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