Theorem rababg 43004

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 547	. . 3 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1814	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		nfv 1910	. . 3 ⊢ Ⅎ𝑦(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfsab1 2713	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5		nfrab1 3448	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
6	5	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
7	4, 6	nfim 1892	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
8		abid 2709	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		eleq1w 2812	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	bitr3id 285	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
11		rabid 3449	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
12		eleq1w 2812	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
13	11, 12	bitr3id 285	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
14	10, 13	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})))
15	3, 7, 14	cbvalv1 2333	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
16		eqss 3995	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
17		rabssab 4081	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
18	17	biantrur 530	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
19		dfss2 3967	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
20	16, 18, 19	3bitr2ri 300	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})
21	2, 15, 20	3bitri 297	1 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   = wceq 1534   ∈ wcel 2099  {cab 2705  {crab 3429   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by: (None)