Theorem rexrsb 44592

Description: An equivalent expression for restricted existence, analogous to exsb 2357. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Assertion

Ref	Expression
rexrsb	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexrsb

Step	Hyp	Ref	Expression
1		rexsb 44591	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		alral 3080	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
3		df-ral 3069	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
4		19.27v 1993	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) ↔ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴))
5		pm2.04 90	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝜑)))
6		eleq1w 2821	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
7	6	biimprd 247	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐴))
8	7	imim1d 82	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) → (𝑦 ∈ 𝐴 → 𝜑)))
9	8	a2i 14	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 → 𝜑)))
10		pm2.04 90	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 → (𝑦 ∈ 𝐴 → 𝜑)) → (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
11	5, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
12	11	imp 407	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 → 𝜑))
13	12	alimi 1814	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
14	4, 13	sylbir 234	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
15	14	ex 413	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
16	3, 15	sylbi 216	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑) → (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
17	16	com12 32	. . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
18	2, 17	impbid2 225	. . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑)))
19	18	rexbiia 3180	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
20	1, 19	bitri 274	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065

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-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070

This theorem is referenced by:  2rexrsb  44594