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Theorem eusvnf 4212
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 1946 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 vex 2577 . . . . . . 7 𝑧 ∈ V
3 nfcv 2194 . . . . . . . 8 𝑥𝑧
4 nfcsb1v 2909 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
54nfeq2 2205 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
6 csbeq1a 2887 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
76eqeq2d 2067 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
83, 5, 7spcgf 2652 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
92, 8ax-mp 7 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
10 vex 2577 . . . . . . 7 𝑤 ∈ V
11 nfcv 2194 . . . . . . . 8 𝑥𝑤
12 nfcsb1v 2909 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1312nfeq2 2205 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
14 csbeq1a 2887 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2067 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1611, 13, 15spcgf 2652 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1710, 16ax-mp 7 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
189, 17eqtr3d 2090 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1918alrimivv 1771 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
20 sbnfc2 2933 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
2119, 20sylibr 141 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2221exlimiv 1505 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
231, 22syl 14 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wex 1397  wcel 1409  ∃!weu 1916  wnfc 2181  Vcvv 2574  csb 2879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787  df-csb 2880
This theorem is referenced by:  eusvnfb  4213  eusv2i  4214
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