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Theorem spc2ev 3274
 Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypotheses
Ref Expression
spc2ev.1 𝐴 ∈ V
spc2ev.2 𝐵 ∈ V
spc2ev.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
spc2ev (𝜓 → ∃𝑥𝑦𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem spc2ev
StepHypRef Expression
1 spc2ev.1 . 2 𝐴 ∈ V
2 spc2ev.2 . 2 𝐵 ∈ V
3 spc2ev.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43spc2egv 3268 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥𝑦𝜑))
51, 2, 4mp2an 704 1 (𝜓 → ∃𝑥𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by:  relop  5194  endisj  7932  dcomex  9152  axcnre  9864  constr3cyclpe  26191  3v3e3cycl2  26192  qqhval2  29354  itg2addnclem3  32633  funop1  40327  1wlk2f  40834  uhgr3cyclex  41349
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