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Theorem cgsex4g 3441
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2144, ax-11 2161. (Revised by Gino Giotto, 28-Jun-2024.)
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
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cgsex4g.2 . . . . 5 (𝜒 → (𝜑𝜓))
21biimpa 480 . . . 4 ((𝜒𝜑) → 𝜓)
32exlimivv 1938 . . 3 (∃𝑧𝑤(𝜒𝜑) → 𝜓)
43exlimivv 1938 . 2 (∃𝑥𝑦𝑧𝑤(𝜒𝜑) → 𝜓)
5 elisset 2814 . . . . . . . 8 (𝐴𝑅 → ∃𝑥 𝑥 = 𝐴)
6 elisset 2814 . . . . . . . 8 (𝐵𝑆 → ∃𝑦 𝑦 = 𝐵)
75, 6anim12i 616 . . . . . . 7 ((𝐴𝑅𝐵𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 exdistrv 1962 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
97, 8sylibr 237 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 elisset 2814 . . . . . . . 8 (𝐶𝑅 → ∃𝑧 𝑧 = 𝐶)
11 elisset 2814 . . . . . . . 8 (𝐷𝑆 → ∃𝑤 𝑤 = 𝐷)
1210, 11anim12i 616 . . . . . . 7 ((𝐶𝑅𝐷𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13 exdistrv 1962 . . . . . . 7 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
1412, 13sylibr 237 . . . . . 6 ((𝐶𝑅𝐷𝑆) → ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷))
159, 14anim12i 616 . . . . 5 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
16 eqeq1 2742 . . . . . . . . . . . . . . 15 (𝑧 = 𝑣 → (𝑧 = 𝐶𝑣 = 𝐶))
1716anbi1d 633 . . . . . . . . . . . . . 14 (𝑧 = 𝑣 → ((𝑧 = 𝐶𝑤 = 𝐷) ↔ (𝑣 = 𝐶𝑤 = 𝐷)))
1817anbi2d 632 . . . . . . . . . . . . 13 (𝑧 = 𝑣 → (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑣 = 𝐶𝑤 = 𝐷))))
1918exbidv 1927 . . . . . . . . . . . 12 (𝑧 = 𝑣 → (∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑣 = 𝐶𝑤 = 𝐷))))
2019notbid 321 . . . . . . . . . . 11 (𝑧 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑣 = 𝐶𝑤 = 𝐷))))
2120alcomiw 2054 . . . . . . . . . 10 (∀𝑦𝑧 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∀𝑧𝑦 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
22 eqeq1 2742 . . . . . . . . . . . . . . 15 (𝑦 = 𝑣 → (𝑦 = 𝐵𝑣 = 𝐵))
2322anbi2d 632 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝑣 = 𝐵)))
2423anbi1d 633 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴𝑣 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷))))
2524exbidv 1927 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴𝑣 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷))))
2625notbid 321 . . . . . . . . . . 11 (𝑦 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴𝑣 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷))))
2726alcomiw 2054 . . . . . . . . . 10 (∀𝑧𝑦 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∀𝑦𝑧 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
2821, 27impbii 212 . . . . . . . . 9 (∀𝑦𝑧 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∀𝑧𝑦 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
2928notbii 323 . . . . . . . 8 (¬ ∀𝑦𝑧 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ¬ ∀𝑧𝑦 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
30 2exnaln 1835 . . . . . . . 8 (∃𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ¬ ∀𝑦𝑧 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
31 2exnaln 1835 . . . . . . . 8 (∃𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ¬ ∀𝑧𝑦 ¬ ∃𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
3229, 30, 313bitr4i 306 . . . . . . 7 (∃𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
3332exbii 1854 . . . . . 6 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑥𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
34 4exdistrv 1963 . . . . . 6 (∃𝑥𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
3533, 34bitri 278 . . . . 5 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
3615, 35sylibr 237 . . . 4 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
37 cgsex4g.1 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
38372eximi 1842 . . . . 5 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑧𝑤𝜒)
39382eximi 1842 . . . 4 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑥𝑦𝑧𝑤𝜒)
4036, 39syl 17 . . 3 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤𝜒)
411biimprcd 253 . . . . . 6 (𝜓 → (𝜒𝜑))
4241ancld 554 . . . . 5 (𝜓 → (𝜒 → (𝜒𝜑)))
43422eximdv 1925 . . . 4 (𝜓 → (∃𝑧𝑤𝜒 → ∃𝑧𝑤(𝜒𝜑)))
44432eximdv 1925 . . 3 (𝜓 → (∃𝑥𝑦𝑧𝑤𝜒 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
4540, 44syl5com 31 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (𝜓 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
464, 45impbid2 229 1 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wex 1786  wcel 2113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811
This theorem is referenced by:  copsex4g  5349  brecop  8414
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