Theorem eusv1 5327

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 2195	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴)
2		sp 2195	. . . 4 ⊢ (∀𝑥 𝑧 = 𝐴 → 𝑧 = 𝐴)
3		eqtr3 2762	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → 𝑦 = 𝑧)
4	1, 2, 3	syl2an 602	. . 3 ⊢ ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
5	4	gen2 1803	. 2 ⊢ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6		eqeq1 2744	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑧 = 𝐴))
7	6	albidv 1927	. . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
8	7	eu4 2619	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (∃𝑦∀𝑥 𝑦 = 𝐴 ∧ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
9	5, 8	mpbiran2 716	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-cleq 2732

This theorem is referenced by:  eusvnfb  5329