Theorem rababg 44328

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 556	. . 3 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1849	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		nfv 1944	. . 3 ⊢ Ⅎ𝑦(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfsab1 2749	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5		nfrab1 3436	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
6	5	nfcri 2917	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
7	4, 6	nfim 1926	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
8		abid 2745	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		eleq1w 2846	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	bitr3id 288	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
11		rabid 3437	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
12		eleq1w 2846	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
13	11, 12	bitr3id 288	. . . 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 2373	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
16		eqss 3952	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
17		rabssab 4039	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
18	17	biantrur 539	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} ∧ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
19		df-ss 3922	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
20	16, 18, 19	3bitr2ri 303	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})
21	2, 15, 20	3bitri 300	1 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {cab 2741  {crab 3416   ⊆ wss 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-ss 3922

This theorem is used by: (None)