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Theorem elrab3t 3330
 Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3332.) (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 2905 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eleq2i 2680 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
3 id 22 . . 3 (𝐴𝐵𝐴𝐵)
4 nfa1 2015 . . . . 5 𝑥𝑥(𝑥 = 𝐴 → (𝜑𝜓))
5 nfv 1830 . . . . 5 𝑥 𝐴𝐵
64, 5nfan 1816 . . . 4 𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵)
7 sp 2041 . . . . . 6 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
8 eleq1 2676 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
98biimparc 503 . . . . . . . . 9 ((𝐴𝐵𝑥 = 𝐴) → 𝑥𝐵)
109biantrurd 528 . . . . . . . 8 ((𝐴𝐵𝑥 = 𝐴) → (𝜑 ↔ (𝑥𝐵𝜑)))
1110bibi1d 332 . . . . . . 7 ((𝐴𝐵𝑥 = 𝐴) → ((𝜑𝜓) ↔ ((𝑥𝐵𝜑) ↔ 𝜓)))
1211pm5.74da 719 . . . . . 6 (𝐴𝐵 → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
137, 12syl5ibcom 234 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
1413imp 444 . . . 4 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
156, 14alrimi 2069 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
16 elabgt 3316 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
173, 15, 16syl2an2 871 . 2 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
182, 17syl5bb 271 1 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175 This theorem is referenced by:  f1oresrab  6302  poimirlem17  32596
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