Theorem elrab3t 2907

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2909.) (Contributed by Thierry Arnoux, 31-Aug-2017.)

Assertion

Ref	Expression
elrab3t	⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab3t

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
2		nfa1 1552	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		nfv 1539	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
4	2, 3	nfan 1576	. . . 4 ⊢ Ⅎ𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵)
5		simpl 109	. . . . . 6 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)))
6	5	19.21bi 1569	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)))
7		eleq1 2252	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
8	7	biimparc 299	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐵)
9	8	biantrurd 305	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
10	9	bibi1d 233	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → ((𝜑 ↔ 𝜓) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
11	10	pm5.74da 443	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))))
12	11	adantl 277	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))))
13	6, 12	mpbid 147	. . . 4 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
14	4, 13	alrimi 1533	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
15		elabgt 2893	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓))
16	1, 14, 15	syl2anc 411	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓))
17		df-rab 2477	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
18	17	eleq2i 2256	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
19	18	bibi1i 228	. 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓) ↔ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓))
20	16, 19	sylibr 134	1 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2160  {cab 2175  {crab 2472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754

This theorem is referenced by:  f1oresrab  5701