Theorem rabeq0w 4410

Description: Condition for a restricted class abstraction to be empty. Version of rabeq0 4411 using implicit substitution, which does not require ax-10 2141, ax-11 2158, ax-12 2178, but requires ax-8 2110. (Contributed by GG, 30-Sep-2024.)

Hypothesis

Ref	Expression
rabeq0w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabeq0w	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rabeq0w

Step	Hyp	Ref	Expression
1		eleq1w 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		rabeq0w.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 631	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4	3	ab0w 4401	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = ∅ ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ 𝜓))
5		df-rab 3444	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	eqeq1i 2745	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = ∅)
7		raln 3075	. 2 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ 𝜓))
8	4, 6, 7	3bitr4i 303	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  {crab 3443  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-dif 3979  df-nul 4353

This theorem is referenced by:  dffr2  5661  frc  5663  frirr  5676