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Theorem elrab3t 3468
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3470.) (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 3023 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eleq2i 2795 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
3 id 22 . . 3 (𝐴𝐵𝐴𝐵)
4 nfa1 2141 . . . . 5 𝑥𝑥(𝑥 = 𝐴 → (𝜑𝜓))
5 nfv 1956 . . . . 5 𝑥 𝐴𝐵
64, 5nfan 1941 . . . 4 𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵)
7 sp 2164 . . . . . 6 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
8 eleq1 2791 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
98biimparc 505 . . . . . . . . 9 ((𝐴𝐵𝑥 = 𝐴) → 𝑥𝐵)
109biantrurd 530 . . . . . . . 8 ((𝐴𝐵𝑥 = 𝐴) → (𝜑 ↔ (𝑥𝐵𝜑)))
1110bibi1d 332 . . . . . . 7 ((𝐴𝐵𝑥 = 𝐴) → ((𝜑𝜓) ↔ ((𝑥𝐵𝜑) ↔ 𝜓)))
1211pm5.74da 725 . . . . . 6 (𝐴𝐵 → ((𝑥 = 𝐴 → (𝜑𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
137, 12syl5ibcom 235 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))))
1413imp 444 . . . 4 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
156, 14alrimi 2193 . . 3 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓)))
16 elabgt 3452 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
173, 15, 16syl2an2 910 . 2 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ 𝜓))
182, 17syl5bb 272 1 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1594   = wceq 1596  wcel 2103  {cab 2710  {crab 3018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306
This theorem is referenced by:  f1oresrab  6510  poimirlem17  33658
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