MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spc2ev Structured version   Visualization version   GIF version

Theorem spc2ev 3594
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 3586 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥𝑦𝜑))
51, 2, 4mp2an 691 1 (𝜓 → ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896
This theorem is referenced by:  relop  5709  endisj  8596  dcomex  9863  axcnre  10580  hashle2pr  13838  wlk2f  27417  uhgr3cyclex  27965  qqhval2  31250  satfv1  32637  itg2addnclem3  35022  funop1  43705
  Copyright terms: Public domain W3C validator