Theorem eusv2 4865

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 4864	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
3		eusvnfb 4862	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
4	1, 3	mpbiran2 954	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
5	2, 4	bitr4i 267	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴)

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  Ⅎwnfc 2751  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)