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Theorem cgsex4g 2656
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
cgsex4g.1 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
cgsex4g.2 (𝜒 → (𝜑𝜓))
Assertion
Ref Expression
cgsex4g (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝜓,𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜒(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦,𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cgsex4g
StepHypRef Expression
1 cgsex4g.2 . . . . 5 (𝜒 → (𝜑𝜓))
21biimpa 290 . . . 4 ((𝜒𝜑) → 𝜓)
32exlimivv 1824 . . 3 (∃𝑧𝑤(𝜒𝜑) → 𝜓)
43exlimivv 1824 . 2 (∃𝑥𝑦𝑧𝑤(𝜒𝜑) → 𝜓)
5 elisset 2633 . . . . . . . 8 (𝐴𝑅 → ∃𝑥 𝑥 = 𝐴)
6 elisset 2633 . . . . . . . 8 (𝐵𝑆 → ∃𝑦 𝑦 = 𝐵)
75, 6anim12i 331 . . . . . . 7 ((𝐴𝑅𝐵𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 eeanv 1855 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
97, 8sylibr 132 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 elisset 2633 . . . . . . . 8 (𝐶𝑅 → ∃𝑧 𝑧 = 𝐶)
11 elisset 2633 . . . . . . . 8 (𝐷𝑆 → ∃𝑤 𝑤 = 𝐷)
1210, 11anim12i 331 . . . . . . 7 ((𝐶𝑅𝐷𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13 eeanv 1855 . . . . . . 7 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
1412, 13sylibr 132 . . . . . 6 ((𝐶𝑅𝐷𝑆) → ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷))
159, 14anim12i 331 . . . . 5 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
16 ee4anv 1857 . . . . 5 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
1715, 16sylibr 132 . . . 4 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
18 cgsex4g.1 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
19182eximi 1537 . . . . 5 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑧𝑤𝜒)
20192eximi 1537 . . . 4 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑥𝑦𝑧𝑤𝜒)
2117, 20syl 14 . . 3 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤𝜒)
221biimprcd 158 . . . . . 6 (𝜓 → (𝜒𝜑))
2322ancld 318 . . . . 5 (𝜓 → (𝜒 → (𝜒𝜑)))
24232eximdv 1810 . . . 4 (𝜓 → (∃𝑧𝑤𝜒 → ∃𝑧𝑤(𝜒𝜑)))
25242eximdv 1810 . . 3 (𝜓 → (∃𝑥𝑦𝑧𝑤𝜒 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
2621, 25syl5com 29 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (𝜓 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
274, 26impbid2 141 1 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  copsex4g  4065  brecop  6362
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