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Theorem spc2egv 3542
Description: Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypothesis
Ref Expression
spc2egv.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
spc2egv ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2egv
StepHypRef Expression
1 elisset 3448 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 3448 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
31, 2anim12i 612 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 exdistrv 1933 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylibr 235 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
6 spc2egv.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimprcd 251 . . 3 (𝜓 → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜑))
872eximdv 1897 . 2 (𝜓 → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜑))
95, 8syl5com 31 1 ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-cleq 2788  df-clel 2863
This theorem is referenced by:  spc2gv  3543  spc3egv  3546  spc2ev  3550  tpres  6830  addsrpr  10343  mulsrpr  10344  2pthon3v  27409  umgr2wlk  27415  0pthonv  27595  1pthon2v  27619  satfv1  32218  sat1el2xp  32234  dvnprodlem1  41772  dfatcolem  42970
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