MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cgsex4g Structured version   Visualization version   GIF version

Theorem cgsex4g 3393
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 468 . . . 4 ((𝜒𝜑) → 𝜓)
32exlimivv 2027 . . 3 (∃𝑧𝑤(𝜒𝜑) → 𝜓)
43exlimivv 2027 . 2 (∃𝑥𝑦𝑧𝑤(𝜒𝜑) → 𝜓)
5 elisset 3368 . . . . . . . 8 (𝐴𝑅 → ∃𝑥 𝑥 = 𝐴)
6 elisset 3368 . . . . . . . 8 (𝐵𝑆 → ∃𝑦 𝑦 = 𝐵)
75, 6anim12i 606 . . . . . . 7 ((𝐴𝑅𝐵𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 eeanv 2346 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
97, 8sylibr 225 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 elisset 3368 . . . . . . . 8 (𝐶𝑅 → ∃𝑧 𝑧 = 𝐶)
11 elisset 3368 . . . . . . . 8 (𝐷𝑆 → ∃𝑤 𝑤 = 𝐷)
1210, 11anim12i 606 . . . . . . 7 ((𝐶𝑅𝐷𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13 eeanv 2346 . . . . . . 7 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
1412, 13sylibr 225 . . . . . 6 ((𝐶𝑅𝐷𝑆) → ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷))
159, 14anim12i 606 . . . . 5 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
16 ee4anv 2348 . . . . 5 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
1715, 16sylibr 225 . . . 4 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
18 cgsex4g.1 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
19182eximi 1930 . . . . 5 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑧𝑤𝜒)
20192eximi 1930 . . . 4 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑥𝑦𝑧𝑤𝜒)
2117, 20syl 17 . . 3 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤𝜒)
221biimprcd 241 . . . . . 6 (𝜓 → (𝜒𝜑))
2322ancld 546 . . . . 5 (𝜓 → (𝜒 → (𝜒𝜑)))
24232eximdv 2014 . . . 4 (𝜓 → (∃𝑧𝑤𝜒 → ∃𝑧𝑤(𝜒𝜑)))
25242eximdv 2014 . . 3 (𝜓 → (∃𝑥𝑦𝑧𝑤𝜒 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
2621, 25syl5com 31 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (𝜓 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
274, 26impbid2 217 1 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352
This theorem is referenced by:  copsex4g  5116  brecop  8047
  Copyright terms: Public domain W3C validator