Theorem elabgt 3596

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3600.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)

Assertion

Ref	Expression
elabgt	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		elab6g 3593	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
2	1	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
3		elisset 2820	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
4		biimp 214	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
5	4	imim3i 64	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)))
6	5	al2imi 1819	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
7		19.23v 1946	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
8	6, 7	syl6ib 250	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	8	com3r 87	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)))
10		biimpr 219	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
11	10	imim2i 16	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
12	11	alimi 1815	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)))
13		bi2.04 388	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 → (𝜓 → 𝜑)) ↔ (𝜓 → (𝑥 = 𝐴 → 𝜑)))
14	13	albii 1823	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) ↔ ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
15		19.21v 1943	. . . . . . . 8 ⊢ (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
16	14, 15	sylbb 218	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
17	12, 16	syl 17	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
18	17	a1i 11	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
19	9, 18	impbidd 209	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)))
20	3, 19	syl 17	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)))
21	20	imp 406	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
22	2, 21	bitrd 278	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817

This theorem is referenced by:  elrab3t  3616  dfrtrcl2  14701  iinabrex  30809  abfmpeld  30893  abfmpel  30894  bj-elgab  35054  dftrcl3  41217  dfrtrcl3  41230