Theorem elabgt 3617

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3621.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by GG, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 11-May-2025.) (Proof shortened by SN, 1-Dec-2025.)

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 3614	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
2		pm5.74 271	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)))
3	2	biimpi 217	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)))
4	3	alimi 1818	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)))
5		albi 1825	. . . 4 ⊢ (∀𝑥((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6	4, 5	syl 17	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)))
7	1, 6	sylan9bb 514	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)))
8		19.23v 1949	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
9		elisset 2822	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
10		pm5.5 362	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
11	9, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((∃𝑥 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
12	8, 11	bitrid 284	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
13	12	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
14	7, 13	bitrd 280	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815

This theorem is referenced by:  elabg  3621  elrab3t  3635  dfrtrcl2  15022  iinabrex  32665  abfmpeld  32753  abfmpel  32754  bj-elgab  37299  dftrcl3  44171  dfrtrcl3  44184