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Theorem eusv4 5075
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 5069 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2 eusv4.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑦𝐴𝐵 ∈ V)
41, 3mprg 3106 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  ∃!weu 2608  wral 3088  wrex 3089  Vcvv 3384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-nul 4982  ax-pow 5034
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-v 3386  df-dif 3771  df-nul 4115
This theorem is referenced by: (None)
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