Theorem eusv2 4545

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 4544	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
3		eusvnfb 4542	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
4	1, 3	mpbiran2 947	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
5	2, 4	bitr4i 187	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   ↔ wb 105  ∀wal 1393   = wceq 1395  ∃wex 1538  ∃!weu 2077   ∈ wcel 2200  Ⅎwnfc 2359  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888

This theorem is referenced by: (None)