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Theorem spc2ev 3606
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 3598 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥𝑦𝜑))
51, 2, 4mp2an 692 1 (𝜓 → ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-clel 2813
This theorem is referenced by:  relop  5863  endisj  9096  dcomex  10484  axcnre  11201  hashle2pr  14512  wlk2f  29662  uhgr3cyclex  30210  qqhval2  33944  satfv1  35347  itg2addnclem3  37659  funop1  47232
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