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Theorem eusv4 5302
 Description: Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.)
Hypothesis
Ref Expression
eusv4.1 𝐵 ∈ V
Assertion
Ref Expression
eusv4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusv4
StepHypRef Expression
1 reusv2lem3 5296 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2 eusv4.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑦𝐴𝐵 ∈ V)
41, 3mprg 3156 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1530   ∈ wcel 2107  ∃!weu 2650  ∀wral 3142  ∃wrex 3143  Vcvv 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206  ax-pow 5262 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-v 3501  df-dif 3942  df-nul 4295 This theorem is referenced by: (None)
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