Theorem reusv2lem1 5298

Description: Lemma for reusv2 5303. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2lem1	⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

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

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem1

Step	Hyp	Ref	Expression
1		n0 4309	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
2		nfra1 3219	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
3	2	nfmov 2640	. . . 4 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵
4		rsp 3205	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → (𝑦 ∈ 𝐴 → 𝑥 = 𝐵))
5	4	com12 32	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
6	5	alrimiv 1924	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥(∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
7		mo2icl 3704	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵) → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
8	6, 7	syl 17	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
9	3, 8	exlimi 2213	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
10	1, 9	sylbi 219	. 2 ⊢ (𝐴 ≠ ∅ → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
11		df-eu 2650	. . 3 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
12	11	rbaib 541	. 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
13	10, 12	syl 17	1 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃*wmo 2616  ∃!weu 2649   ≠ wne 3016  ∀wral 3138  ∅c0 4290

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-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-v 3496  df-dif 3938  df-nul 4291

This theorem is referenced by: (None)