Theorem rabeqcOLD 3673

Description: Obsolete version of rabeqc 3436 as of 15-Jan-2025. (Contributed by AV, 20-Apr-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rabeqcOLD.1	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Assertion

Ref	Expression
rabeqcOLD	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabeqcOLD

Step	Hyp	Ref	Expression
1		df-rab 3425	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		eqabcb 2867	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴))
3		rabeqcOLD.1	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝜑)
4	3	pm4.71i 559	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	bicomi 223	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴)
6	2, 5	mpgbir 1793	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴
7	1, 6	eqtri 2752	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  {crab 3424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425

This theorem is referenced by: (None)