Theorem reusv2lem1 4868

Description: Lemma for reusv2 4874. (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 3931	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
2		nfra1 2941	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
3	2	nfmo 2487	. . . 4 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵
4		rsp 2929	. . . . . . 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 1855	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥(∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
7		mo2icl 3385	. . . . 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 2086	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
10	1, 9	sylbi 207	. 2 ⊢ (𝐴 ≠ ∅ → ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
11		eu5 2496	. . 3 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
12	11	rbaib 947	. 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 196  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471   ≠ wne 2794  ∀wral 2912  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)