Theorem eusv1 4453

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 1511	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)
2		sp 1511	. . . 4 ⊢ (∀𝑥 𝑧 = 𝐴 → 𝑧 = 𝐴)
3		eqtr3 2197	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → 𝑦 = 𝑧)
4	1, 2, 3	syl2an 289	. . 3 ⊢ ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
5	4	gen2 1450	. 2 ⊢ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6		eqeq1 2184	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑧 = 𝐴))
7	6	albidv 1824	. . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
8	7	eu4 2088	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (∃𝑦∀𝑥 𝑦 = 𝐴 ∧ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
9	5, 8	mpbiran2 941	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492  ∃!weu 2026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170

This theorem is referenced by:  eusvnfb  4455