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Theorem elabgt 3671
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3675.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by GG, 12-Oct-2024.) (Proof shortend by Wolf Lammen, 11-May-2025.)
Assertion
Ref Expression
elabgt ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elabgt
StepHypRef Expression
1 elab6g 3668 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
21adantr 480 . 2 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
3 elisset 2822 . . . . . 6 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
4 biimp 215 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
54imim3i 64 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓)))
65al2imi 1814 . . . . . . 7 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → ∀𝑥(𝑥 = 𝐴𝜓)))
7 19.23v 1941 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓))
86, 7imbitrdi 251 . . . . . 6 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓)))
93, 8syl7 74 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → (𝐴𝐵𝜓)))
109com3r 87 . . . 4 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓)))
1110imp 406 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
12 biimpr 220 . . . . . . . 8 ((𝜑𝜓) → (𝜓𝜑))
1312imim2i 16 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1413com23 86 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1514alimi 1810 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
16 19.21v 1938 . . . . 5 (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1715, 16sylib 218 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1817adantl 481 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1911, 18impbid 212 . 2 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
202, 19bitrd 279 1 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107  {cab 2713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815
This theorem is referenced by:  elrab3t  3690  dfrtrcl2  15102  iinabrex  32583  abfmpeld  32665  abfmpel  32666  bj-elgab  36941  dftrcl3  43738  dfrtrcl3  43751
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