Theorem eusv2 4207

Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypothesis

Ref	Expression
eusv2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eusv2	⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2

Step	Hyp	Ref	Expression
1		eusv2.1	. . 3 ⊢ 𝐴 ∈ V
2	1	eusv2nf 4206	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
3		eusvnfb 4204	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
4	1, 3	mpbiran2 882	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
5	2, 4	bitr4i 185	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃!weu 1941  Ⅎwnfc 2206  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602

This theorem is referenced by: (None)