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Theorem elrab3t 2749
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2751.) (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 108 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → 𝐴𝐵)
2 nfa1 1475 . . . . 5 𝑥𝑥(𝑥 = 𝐴 → (𝜑𝜓))
3 nfv 1462 . . . . 5 𝑥 𝐴𝐵
42, 3nfan 1498 . . . 4 𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵)
5 simpl 107 . . . . . 6 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)))
6519.21bi 1491 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → (𝜑𝜓)))
7 eleq1 2142 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
87biimparc 293 . . . . . . . . 9 ((𝐴𝐵𝑥 = 𝐴) → 𝑥𝐵)
98biantrurd 299 . . . . . . . 8 ((𝐴𝐵𝑥 = 𝐴) → (𝜑 ↔ (𝑥𝐵𝜑)))
109bibi1d 231 . . . . . . 7 ((𝐴𝐵𝑥 = 𝐴) → ((𝜑𝜓) ↔ ((𝑥𝐵𝜑) ↔ 𝜓)))
1110pm5.74da 432 . . . . . 6 (𝐴𝐵 → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
1211adantl 271 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
136, 12mpbid 145 . . . 4 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
144, 13alrimi 1456 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
15 elabgt 2736 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
161, 14, 15syl2anc 403 . 2 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
17 df-rab 2358 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
1817eleq2i 2146 . . 3 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
1918bibi1i 226 . 2 ((𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓) ↔ (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
2016, 19sylibr 132 1 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2068  {crab 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604
This theorem is referenced by:  f1oresrab  5361
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