Theorem elrab3t 3678

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3680.) (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		df-rab 3147	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	eleq2i 2904	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵)
4		nfa1 2151	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5		nfv 1911	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
6	4, 5	nfan 1896	. . . 4 ⊢ Ⅎ𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵)
7		sp 2178	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)))
8		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
9	8	biimparc 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐵)
10	9	biantrurd 535	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
11	10	bibi1d 346	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → ((𝜑 ↔ 𝜓) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
12	11	pm5.74da 802	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))))
13	7, 12	syl5ibcom 247	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))))
14	13	imp 409	. . . 4 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
15	6, 14	alrimi 2209	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))
16		elabgt 3662	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓))
17	3, 15, 16	syl2an2 684	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓))
18	2, 17	syl5bb 285	1 ⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  {cab 2799  {crab 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496

This theorem is referenced by:  f1oresrab  6883  poimirlem17  34903