Theorem rabeqcOLD 3680

Description: Obsolete version of rabeqc 3444 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 3433	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		eqabcb 2875	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴))
3		rabeqcOLD.1	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝜑)
4	3	pm4.71i 560	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	bicomi 223	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴)
6	2, 5	mpgbir 1801	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴
7	1, 6	eqtri 2760	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  {crab 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433

This theorem is referenced by: (None)