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Theorem spc2egv 3597
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 3503 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 3503 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
31, 2anim12i 612 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 exdistrv 1947 . . 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 1911 . 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 1528  wex 1771  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890
This theorem is referenced by:  spc2gv  3598  spc3egv  3601  spc2ev  3605  tpres  6955  addsrpr  10485  mulsrpr  10486  2pthon3v  27649  umgr2wlk  27655  0pthonv  27835  1pthon2v  27859  satfv1  32507  sat1el2xp  32523  dvnprodlem1  42107  dfatcolem  43331  fundcmpsurbijinj  43447
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