Theorem rexrsb 47050

Description: An equivalent expression for restricted existence, analogous to exsb 2360. (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 47049	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		alral 3073	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
3		df-ral 3060	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
4		19.27v 1987	. . . . . . . 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 2822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
7	6	biimprd 248	. . . . . . . . . . . . 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 406	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 → 𝜑))
13	12	alimi 1808	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
14	4, 13	sylbir 235	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ∧ 𝑦 ∈ 𝐴) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
15	14	ex 412	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
16	3, 15	sylbi 217	. . . . 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 226	. . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑)))
19	18	rexbiia 3090	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
20	1, 19	bitri 275	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069

This theorem is referenced by:  2rexrsb  47052