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Theorem spc2ed 5885
Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
spc2ed.x 𝑥𝜒
spc2ed.y 𝑦𝜒
spc2ed.1 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
spc2ed ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝜒 → ∃𝑥𝑦𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2ed
StepHypRef Expression
1 elisset 2614 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2614 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
31, 2anim12i 331 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 eeanv 1849 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylibr 132 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
6 nfv 1462 . . . . 5 𝑥𝜑
7 spc2ed.x . . . . 5 𝑥𝜒
86, 7nfan 1498 . . . 4 𝑥(𝜑𝜒)
9 nfv 1462 . . . . . 6 𝑦𝜑
10 spc2ed.y . . . . . 6 𝑦𝜒
119, 10nfan 1498 . . . . 5 𝑦(𝜑𝜒)
12 anass 393 . . . . . . . 8 (((𝜒𝜑) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) ↔ (𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵))))
13 ancom 262 . . . . . . . . 9 ((𝜒𝜑) ↔ (𝜑𝜒))
1413anbi1i 446 . . . . . . . 8 (((𝜒𝜑) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) ↔ ((𝜑𝜒) ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
1512, 14bitr3i 184 . . . . . . 7 ((𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵))) ↔ ((𝜑𝜒) ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
16 spc2ed.1 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
1716biimparc 293 . . . . . . 7 ((𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵))) → 𝜓)
1815, 17sylbir 133 . . . . . 6 (((𝜑𝜒) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝜓)
1918ex 113 . . . . 5 ((𝜑𝜒) → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜓))
2011, 19eximd 1544 . . . 4 ((𝜑𝜒) → (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑦𝜓))
218, 20eximd 1544 . . 3 ((𝜑𝜒) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜓))
2221impancom 256 . 2 ((𝜑 ∧ ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵)) → (𝜒 → ∃𝑥𝑦𝜓))
235, 22sylan2 280 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝜒 → ∃𝑥𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wex 1422  wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  cnvoprab  5886
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