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Theorem eusvnf 4471
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 2068 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 vex 2755 . . . . . . 7 𝑧 ∈ V
3 nfcv 2332 . . . . . . . 8 𝑥𝑧
4 nfcsb1v 3105 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
54nfeq2 2344 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
6 csbeq1a 3081 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
76eqeq2d 2201 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
83, 5, 7spcgf 2834 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
92, 8ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
10 vex 2755 . . . . . . 7 𝑤 ∈ V
11 nfcv 2332 . . . . . . . 8 𝑥𝑤
12 nfcsb1v 3105 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1312nfeq2 2344 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
14 csbeq1a 3081 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2201 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1611, 13, 15spcgf 2834 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1710, 16ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
189, 17eqtr3d 2224 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1918alrimivv 1886 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
20 sbnfc2 3132 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
2119, 20sylibr 134 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2221exlimiv 1609 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
231, 22syl 14 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wex 1503  ∃!weu 2038  wcel 2160  wnfc 2319  Vcvv 2752  csb 3072
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  eusvnfb  4472  eusv2i  4473
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