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Theorem eusvnf 5029
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 2591 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 vex 3353 . . . . . . 7 𝑧 ∈ V
3 nfcv 2907 . . . . . . . 8 𝑥𝑧
4 nfcsb1v 3709 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
54nfeq2 2923 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
6 csbeq1a 3702 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
76eqeq2d 2775 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
83, 5, 7spcgf 3441 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
92, 8ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
10 vex 3353 . . . . . . 7 𝑤 ∈ V
11 nfcv 2907 . . . . . . . 8 𝑥𝑤
12 nfcsb1v 3709 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1312nfeq2 2923 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
14 csbeq1a 3702 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2775 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1611, 13, 15spcgf 3441 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1710, 16ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
189, 17eqtr3d 2801 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1918alrimivv 2023 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
20 sbnfc2 4171 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
2119, 20sylibr 225 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2221exlimiv 2025 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
231, 22syl 17 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  wnfc 2894  Vcvv 3350  csb 3693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-nul 4082
This theorem is referenced by:  eusvnfb  5030  eusv2i  5031
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