MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eusvnf Structured version   Visualization version   GIF version

Theorem eusvnf 5142
Description: Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
eusvnf (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusvnf
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 euex 2597 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 nfcv 2925 . . . . . . . 8 𝑥𝑧
3 nfcsb1v 3797 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
43nfeq2 2940 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
5 csbeq1a 3788 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
65eqeq2d 2781 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
72, 4, 6spcgf 3502 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
87elv 3413 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
9 nfcv 2925 . . . . . . . 8 𝑥𝑤
10 nfcsb1v 3797 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1110nfeq2 2940 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
12 csbeq1a 3788 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1312eqeq2d 2781 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
149, 11, 13spcgf 3502 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1514elv 3413 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
168, 15eqtr3d 2809 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1716alrimivv 1888 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
18 sbnfc2 4266 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1917, 18sylibr 226 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2019exlimiv 1890 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
211, 20syl 17 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1506   = wceq 1508  wex 1743  ∃!weu 2584  wnfc 2909  Vcvv 3408  csb 3779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-nul 4173
This theorem is referenced by:  eusvnfb  5143  eusv2i  5144
  Copyright terms: Public domain W3C validator