Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rababg Structured version   Visualization version   GIF version

Theorem rababg 37399
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 572 . . 3 ((𝜑𝑥𝐴) ↔ (𝜑 → (𝑥𝐴𝜑)))
21albii 1744 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜑)))
3 nfv 1840 . . 3 𝑦(𝜑 → (𝑥𝐴𝜑))
4 nfsab1 2611 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
5 nfrab1 3115 . . . . 5 𝑥{𝑥𝐴𝜑}
65nfcri 2755 . . . 4 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
74, 6nfim 1822 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})
8 abid 2609 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 eleq1 2686 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
108, 9syl5bbr 274 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
11 rabid 3110 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
12 eleq1 2686 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1311, 12syl5bbr 274 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
1410, 13imbi12d 334 . . 3 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐴𝜑)) ↔ (𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑})))
153, 7, 14cbval 2270 . 2 (∀𝑥(𝜑 → (𝑥𝐴𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
16 eqss 3603 . . 3 ({𝑥𝐴𝜑} = {𝑥𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
17 rabssab 3674 . . . 4 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
1817biantrur 527 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ({𝑥𝐴𝜑} ⊆ {𝑥𝜑} ∧ {𝑥𝜑} ⊆ {𝑥𝐴𝜑}))
19 dfss2 3577 . . 3 ({𝑥𝜑} ⊆ {𝑥𝐴𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}))
2016, 18, 193bitr2ri 289 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝑦 ∈ {𝑥𝐴𝜑}) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
212, 15, 203bitri 286 1 (∀𝑥(𝜑𝑥𝐴) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  {crab 2912  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-in 3567  df-ss 3574
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator