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Theorem eusvnf 4826
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 2493 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
2 vex 3192 . . . . . . 7 𝑧 ∈ V
3 nfcv 2761 . . . . . . . 8 𝑥𝑧
4 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
54nfeq2 2776 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐴
6 csbeq1a 3527 . . . . . . . . 9 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
76eqeq2d 2631 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
83, 5, 7spcgf 3277 . . . . . . 7 (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴))
92, 8ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑧 / 𝑥𝐴)
10 vex 3192 . . . . . . 7 𝑤 ∈ V
11 nfcv 2761 . . . . . . . 8 𝑥𝑤
12 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑤 / 𝑥𝐴
1312nfeq2 2776 . . . . . . . 8 𝑥 𝑦 = 𝑤 / 𝑥𝐴
14 csbeq1a 3527 . . . . . . . . 9 (𝑥 = 𝑤𝐴 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2631 . . . . . . . 8 (𝑥 = 𝑤 → (𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1611, 13, 15spcgf 3277 . . . . . . 7 (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴))
1710, 16ax-mp 5 . . . . . 6 (∀𝑥 𝑦 = 𝐴𝑦 = 𝑤 / 𝑥𝐴)
189, 17eqtr3d 2657 . . . . 5 (∀𝑥 𝑦 = 𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1918alrimivv 1853 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
20 sbnfc2 3984 . . . 4 (𝑥𝐴 ↔ ∀𝑧𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
2119, 20sylibr 224 . . 3 (∀𝑥 𝑦 = 𝐴𝑥𝐴)
2221exlimiv 1855 . 2 (∃𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
231, 22syl 17 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  wnfc 2748  Vcvv 3189  csb 3518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-nul 3897
This theorem is referenced by:  eusvnfb  4827  eusv2i  4828
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