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Theorem spc2egv 3268
 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 3188 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 3188 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
31, 2anim12i 588 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 eeanv 2170 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylibr 223 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
6 spc2egv.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimprcd 239 . . 3 (𝜓 → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜑))
872eximdv 1835 . 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977 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:  spc2gv  3269  spc2ev  3274  tpres  6371  addsrpr  9775  mulsrpr  9776  0pthonv  26111  1pthon2v  26123  2pthon3v  26134  usg2wlk  26145  usg2wlkon  26146  dvnprodlem1  38836  2pthon3v-av  41150  umgr2wlk  41156  0pthonv-av  41297  1pthon2v-av  41320
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