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Theorem cgsex4g 3539
 Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2141. (Revised by Gino Giotto, 20-Aug-2023.)
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 479 . . . 4 ((𝜒𝜑) → 𝜓)
32exlimivv 1929 . . 3 (∃𝑧𝑤(𝜒𝜑) → 𝜓)
43exlimivv 1929 . 2 (∃𝑥𝑦𝑧𝑤(𝜒𝜑) → 𝜓)
5 elisset 3505 . . . . . . . 8 (𝐴𝑅 → ∃𝑥 𝑥 = 𝐴)
6 elisset 3505 . . . . . . . 8 (𝐵𝑆 → ∃𝑦 𝑦 = 𝐵)
75, 6anim12i 614 . . . . . . 7 ((𝐴𝑅𝐵𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 exdistrv 1952 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
97, 8sylibr 236 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 elisset 3505 . . . . . . . 8 (𝐶𝑅 → ∃𝑧 𝑧 = 𝐶)
11 elisset 3505 . . . . . . . 8 (𝐷𝑆 → ∃𝑤 𝑤 = 𝐷)
1210, 11anim12i 614 . . . . . . 7 ((𝐶𝑅𝐷𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13 exdistrv 1952 . . . . . . 7 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
1412, 13sylibr 236 . . . . . 6 ((𝐶𝑅𝐷𝑆) → ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷))
159, 14anim12i 614 . . . . 5 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
16 excom 2165 . . . . . . 7 (∃𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
1716exbii 1844 . . . . . 6 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑥𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
18 4exdistrv 1953 . . . . . 6 (∃𝑥𝑧𝑦𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
1917, 18bitri 277 . . . . 5 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
2015, 19sylibr 236 . . . 4 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
21 cgsex4g.1 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
22212eximi 1832 . . . . 5 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑧𝑤𝜒)
23222eximi 1832 . . . 4 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑥𝑦𝑧𝑤𝜒)
2420, 23syl 17 . . 3 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤𝜒)
251biimprcd 252 . . . . . 6 (𝜓 → (𝜒𝜑))
2625ancld 553 . . . . 5 (𝜓 → (𝜒 → (𝜒𝜑)))
27262eximdv 1916 . . . 4 (𝜓 → (∃𝑧𝑤𝜒 → ∃𝑧𝑤(𝜒𝜑)))
28272eximdv 1916 . . 3 (𝜓 → (∃𝑥𝑦𝑧𝑤𝜒 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
2924, 28syl5com 31 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (𝜓 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
304, 29impbid2 228 1 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-11 2157  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893 This theorem is referenced by:  copsex4g  5384  brecop  8389
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