Theorem rababg 39926

Description: Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
rababg	⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})

Proof of Theorem rababg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancrb 550	. . 3 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1816	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		nfv 1911	. . 3 ⊢ Ⅎ𝑦(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfsab1 2808	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5		nfrab1 3385	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
6	5	nfcri 2971	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
7	4, 6	nfim 1893	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
8		abid 2803	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		eleq1w 2895	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	syl5bbr 287	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
11		rabid 3379	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
12		eleq1w 2895	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
13	11, 12	syl5bbr 287	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
14	10, 13	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})))
15	3, 7, 14	cbvalv1 2357	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
16		eqss 3982	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
17		rabssab 4060	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
18	17	biantrur 533	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
19		dfss2 3955	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
20	16, 18, 19	3bitr2ri 302	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})
21	2, 15, 20	3bitri 299	1 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  {cab 2799  {crab 3142   ⊆ wss 3936

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-in 3943  df-ss 3952

This theorem is referenced by: (None)