MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrab3t Structured version   Visualization version   GIF version

Theorem elrab3t 3630
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3632.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
elrab3t ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab3t
StepHypRef Expression
1 df-rab 3118 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eleq2i 2884 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
3 id 22 . . 3 (𝐴𝐵𝐴𝐵)
4 nfa1 2153 . . . . 5 𝑥𝑥(𝑥 = 𝐴 → (𝜑𝜓))
5 nfv 1915 . . . . 5 𝑥 𝐴𝐵
64, 5nfan 1900 . . . 4 𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵)
7 sp 2181 . . . . . 6 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
8 eleq1 2880 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
98biimparc 483 . . . . . . . . 9 ((𝐴𝐵𝑥 = 𝐴) → 𝑥𝐵)
109biantrurd 536 . . . . . . . 8 ((𝐴𝐵𝑥 = 𝐴) → (𝜑 ↔ (𝑥𝐵𝜑)))
1110bibi1d 347 . . . . . . 7 ((𝐴𝐵𝑥 = 𝐴) → ((𝜑𝜓) ↔ ((𝑥𝐵𝜑) ↔ 𝜓)))
1211pm5.74da 803 . . . . . 6 (𝐴𝐵 → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
137, 12syl5ibcom 248 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
1413imp 410 . . . 4 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
156, 14alrimi 2212 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
16 elabgt 3612 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
173, 15, 16syl2an2 685 . 2 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
182, 17syl5bb 286 1 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  {cab 2779  {crab 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446
This theorem is referenced by:  f1oresrab  6870  poimirlem17  35067
  Copyright terms: Public domain W3C validator