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Theorem eusvnf 4266
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 1978 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 vex 2622 . . . . . . 7 𝑧 ∈ V
3 nfcv 2228 . . . . . . . 8 𝑥𝑧
4 nfcsb1v 2961 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
54nfeq2 2240 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
6 csbeq1a 2939 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
76eqeq2d 2099 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
83, 5, 7spcgf 2701 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
92, 8ax-mp 7 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
10 vex 2622 . . . . . . 7 𝑤 ∈ V
11 nfcv 2228 . . . . . . . 8 𝑥𝑤
12 nfcsb1v 2961 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1312nfeq2 2240 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
14 csbeq1a 2939 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2099 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1611, 13, 15spcgf 2701 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1710, 16ax-mp 7 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
189, 17eqtr3d 2122 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1918alrimivv 1803 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
20 sbnfc2 2986 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
2119, 20sylibr 132 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2221exlimiv 1534 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
231, 22syl 14 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  wnfc 2215  Vcvv 2619  csb 2931
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by:  eusvnfb  4267  eusv2i  4268
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