Theorem eusv1 4430

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

Assertion

Ref	Expression
eusv1	⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 1499	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)
2		sp 1499	. . . 4 ⊢ (∀𝑥 𝑧 = 𝐴 → 𝑧 = 𝐴)
3		eqtr3 2185	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → 𝑦 = 𝑧)
4	1, 2, 3	syl2an 287	. . 3 ⊢ ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
5	4	gen2 1438	. 2 ⊢ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6		eqeq1 2172	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑧 = 𝐴))
7	6	albidv 1812	. . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
8	7	eu4 2076	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (∃𝑦∀𝑥 𝑦 = 𝐴 ∧ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
9	5, 8	mpbiran2 931	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480  ∃!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-cleq 2158

This theorem is referenced by:  eusvnfb  4432