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Theorem cgsex4g 3520
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2137, ax-11 2154. (Revised by Gino Giotto, 28-Jun-2024.) Avoid ax-9 2116, ax-ext 2703. (Revised by Wolf Lammen, 21-Mar-2025.)
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 477 . . . 4 ((𝜒𝜑) → 𝜓)
32exlimivv 1935 . . 3 (∃𝑧𝑤(𝜒𝜑) → 𝜓)
43exlimivv 1935 . 2 (∃𝑥𝑦𝑧𝑤(𝜒𝜑) → 𝜓)
5 elisset 2815 . . . . . . 7 (𝐴𝑅 → ∃𝑥 𝑥 = 𝐴)
6 elisset 2815 . . . . . . 7 (𝐵𝑆 → ∃𝑦 𝑦 = 𝐵)
75, 6anim12i 613 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 elisset 2815 . . . . . . 7 (𝐶𝑅 → ∃𝑧 𝑧 = 𝐶)
9 elisset 2815 . . . . . . 7 (𝐷𝑆 → ∃𝑤 𝑤 = 𝐷)
108, 9anim12i 613 . . . . . 6 ((𝐶𝑅𝐷𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
117, 10anim12i 613 . . . . 5 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) ∧ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷)))
12 19.42vv 1961 . . . . . . 7 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
13122exbii 1851 . . . . . 6 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ∃𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
14 19.41vv 1954 . . . . . 6 (∃𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)) ↔ (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)))
15 exdistrv 1959 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
16 exdistrv 1959 . . . . . . 7 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
1715, 16anbi12i 627 . . . . . 6 ((∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷)) ↔ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) ∧ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷)))
1813, 14, 173bitri 296 . . . . 5 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ↔ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) ∧ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷)))
1911, 18sylibr 233 . . . 4 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)))
20 cgsex4g.1 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)
21202eximi 1838 . . . . 5 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑧𝑤𝜒)
22212eximi 1838 . . . 4 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → ∃𝑥𝑦𝑧𝑤𝜒)
2319, 22syl 17 . . 3 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → ∃𝑥𝑦𝑧𝑤𝜒)
241biimprcd 249 . . . . . 6 (𝜓 → (𝜒𝜑))
2524ancld 551 . . . . 5 (𝜓 → (𝜒 → (𝜒𝜑)))
26252eximdv 1922 . . . 4 (𝜓 → (∃𝑧𝑤𝜒 → ∃𝑧𝑤(𝜒𝜑)))
27262eximdv 1922 . . 3 (𝜓 → (∃𝑥𝑦𝑧𝑤𝜒 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
2823, 27syl5com 31 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (𝜓 → ∃𝑥𝑦𝑧𝑤(𝜒𝜑)))
294, 28impbid2 225 1 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-clel 2810
This theorem is referenced by:  copsex4g  5494  brecop  8800
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