Theorem eusv2i 5370

Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusv2i	⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2i

Step	Hyp	Ref	Expression
1		nfeu1 2620	. . 3 ⊢ Ⅎ𝑦∃!𝑦∀𝑥 𝑦 = 𝐴
2		nfcvd 2929	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝑦)
3		eusvnf 5368	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
4	2, 3	nfeqd 2938	. . . . 5 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5	4	nfrd 1824	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6		19.2 2009	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	5, 6	impbid1 228	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
8	1, 7	eubid 2618	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴))
9	8	ibir 271	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃!weu 2599

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-nul 4290

This theorem is used by:  eusv2nf  5371