Theorem bj-elgab 35127

Description: Elements of a generalized class abstraction. (Contributed by BJ, 4-Oct-2024.)

Hypotheses

Ref	Expression
bj-elgab.nf	⊢ (𝜑 → ∀𝑥𝜑)
bj-elgab.nfa	⊢ (𝜑 → Ⅎ𝑥𝐴)
bj-elgab.ex	⊢ (𝜑 → 𝐴 ∈ 𝑉)
bj-elgab.is	⊢ (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒))

Assertion

Ref	Expression
bj-elgab	⊢ (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒))

Proof of Theorem bj-elgab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-gab 35122	. . 3 ⊢ {𝐵 ∣ 𝑥 ∣ 𝜓} = {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)}
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)})
3		bj-elgab.ex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		bj-elgab.nf	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥𝜑)
5	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ∀𝑥𝜑)
6		bj-elgab.nfa	. . . . . . . . . 10 ⊢ (𝜑 → Ⅎ𝑥𝐴)
7		nfcvd 2908	. . . . . . . . . . . 12 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
8		id 22	. . . . . . . . . . . 12 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
9	7, 8	nfeqd 2917	. . . . . . . . . . 11 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
10	9	nf5rd 2189	. . . . . . . . . 10 ⊢ (Ⅎ𝑥𝐴 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
11	6, 10	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
12	11	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ∀𝑥 𝑦 = 𝐴)
13		19.26 1873	. . . . . . . 8 ⊢ (∀𝑥(𝜑 ∧ 𝑦 = 𝐴) ↔ (∀𝑥𝜑 ∧ ∀𝑥 𝑦 = 𝐴))
14	5, 12, 13	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ∀𝑥(𝜑 ∧ 𝑦 = 𝐴))
15		eqeq2 2750	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
16		eqcom 2745	. . . . . . . . . 10 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
17	15, 16	bitrdi 287	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐴 = 𝐵))
18	17	anbi1d 630	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝐵 = 𝑦 ∧ 𝜓) ↔ (𝐴 = 𝐵 ∧ 𝜓)))
19	18	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝐵 = 𝑦 ∧ 𝜓) ↔ (𝐴 = 𝐵 ∧ 𝜓)))
20	14, 19	exbidh 1870	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (∃𝑥(𝐵 = 𝑦 ∧ 𝜓) ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓)))
21	20	ex 413	. . . . 5 ⊢ (𝜑 → (𝑦 = 𝐴 → (∃𝑥(𝐵 = 𝑦 ∧ 𝜓) ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓))))
22	21	alrimiv 1930	. . . 4 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝐴 → (∃𝑥(𝐵 = 𝑦 ∧ 𝜓) ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓))))
23		elabgt 3603	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦(𝑦 = 𝐴 → (∃𝑥(𝐵 = 𝑦 ∧ 𝜓) ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓)))) → (𝐴 ∈ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)} ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓)))
24	3, 22, 23	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 ∈ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)} ↔ ∃𝑥(𝐴 = 𝐵 ∧ 𝜓)))
25		bj-elgab.is	. . 3 ⊢ (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒))
26	24, 25	bitrd 278	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)} ↔ 𝜒))
27	2, 26	syl5bb 283	1 ⊢ (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  {bj-cgab 35121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-bj-gab 35122

This theorem is referenced by:  bj-gabima  35128