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Theorem rababg 41134
Description: Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
rababg (∀𝑥(𝜑𝑥𝐴) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})

Proof of Theorem rababg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ancrb 547 . . 3 ((𝜑𝑥𝐴) ↔ (𝜑 → (𝑥𝐴𝜑)))
21albii 1825 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜑)))
3 nfv 1920 . . 3 𝑦(𝜑 → (𝑥𝐴𝜑))
4 nfsab1 2724 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
5 nfrab1 3315 . . . . 5 𝑥{𝑥𝐴𝜑}
65nfcri 2895 . . . 4 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
74, 6nfim 1902 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})
8 abid 2720 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
108, 9bitr3id 284 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
11 rabid 3308 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
12 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1311, 12bitr3id 284 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1410, 13imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐴𝜑)) ↔ (𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})))
153, 7, 14cbvalv1 2341 . 2 (∀𝑥(𝜑 → (𝑥𝐴𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
16 eqss 3940 . . 3 ({𝑥𝐴𝜑} = {𝑥𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
17 rabssab 4022 . . . 4 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
1817biantrur 530 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
19 dfss2 3911 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
2016, 18, 193bitr2ri 299 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
212, 15, 203bitri 296 1 (∀𝑥(𝜑𝑥𝐴) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1539   = wceq 1541  wcel 2109  {cab 2716  {crab 3069  wss 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3432  df-in 3898  df-ss 3908
This theorem is referenced by: (None)
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