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Theorem rababg 43564
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 1816 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜑)))
3 nfv 1912 . . 3 𝑦(𝜑 → (𝑥𝐴𝜑))
4 nfsab1 2720 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
5 nfrab1 3454 . . . . 5 𝑥{𝑥𝐴𝜑}
65nfcri 2895 . . . 4 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
74, 6nfim 1894 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})
8 abid 2716 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
108, 9bitr3id 285 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
11 rabid 3455 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
12 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1311, 12bitr3id 285 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1410, 13imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐴𝜑)) ↔ (𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})))
153, 7, 14cbvalv1 2342 . 2 (∀𝑥(𝜑 → (𝑥𝐴𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
16 eqss 4011 . . 3 ({𝑥𝐴𝜑} = {𝑥𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
17 rabssab 4095 . . . 4 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
1817biantrur 530 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
19 df-ss 3980 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
2016, 18, 193bitr2ri 300 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
212, 15, 203bitri 297 1 (∀𝑥(𝜑𝑥𝐴) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  wss 3963
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-ss 3980
This theorem is referenced by: (None)
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