ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrab3t GIF version

Theorem elrab3t 2842
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2844.) (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 simpr 109 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → 𝐴𝐵)
2 nfa1 1522 . . . . 5 𝑥𝑥(𝑥 = 𝐴 → (𝜑𝜓))
3 nfv 1509 . . . . 5 𝑥 𝐴𝐵
42, 3nfan 1545 . . . 4 𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵)
5 simpl 108 . . . . . 6 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)))
6519.21bi 1538 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → (𝜑𝜓)))
7 eleq1 2203 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
87biimparc 297 . . . . . . . . 9 ((𝐴𝐵𝑥 = 𝐴) → 𝑥𝐵)
98biantrurd 303 . . . . . . . 8 ((𝐴𝐵𝑥 = 𝐴) → (𝜑 ↔ (𝑥𝐵𝜑)))
109bibi1d 232 . . . . . . 7 ((𝐴𝐵𝑥 = 𝐴) → ((𝜑𝜓) ↔ ((𝑥𝐵𝜑) ↔ 𝜓)))
1110pm5.74da 440 . . . . . 6 (𝐴𝐵 → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
1211adantl 275 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
136, 12mpbid 146 . . . 4 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
144, 13alrimi 1503 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
15 elabgt 2828 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
161, 14, 15syl2anc 409 . 2 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
17 df-rab 2426 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
1817eleq2i 2207 . . 3 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
1918bibi1i 227 . 2 ((𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓) ↔ (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
2016, 19sylibr 133 1 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330   = wceq 1332  wcel 1481  {cab 2126  {crab 2421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691
This theorem is referenced by:  f1oresrab  5592
  Copyright terms: Public domain W3C validator