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Theorem eusvnf 5261
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 2640 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 nfcv 2958 . . . . . . . 8 𝑥𝑧
3 nfcsb1v 3855 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
43nfeq2 2975 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
5 csbeq1a 3845 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
65eqeq2d 2812 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
72, 4, 6spcgf 3541 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
87elv 3449 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
9 nfcv 2958 . . . . . . . 8 𝑥𝑤
10 nfcsb1v 3855 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1110nfeq2 2975 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
12 csbeq1a 3845 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1312eqeq2d 2812 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
149, 11, 13spcgf 3541 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1514elv 3449 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
168, 15eqtr3d 2838 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1716alrimivv 1929 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
18 sbnfc2 4347 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1917, 18sylibr 237 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2019exlimiv 1931 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
211, 20syl 17 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wex 1781  ∃!weu 2631  wnfc 2939  Vcvv 3444  csb 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-nul 4247
This theorem is referenced by:  eusvnfb  5262  eusv2i  5263
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