Theorem rabeqcOLD 3696

Description: Obsolete version of rabeqc 3449 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 3437	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		eqabcb 2883	. . 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 224	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴)
6	2, 5	mpgbir 1798	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴
7	1, 6	eqtri 2765	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2714  {crab 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  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-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437

This theorem is referenced by: (None)