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Theorem cgsex2g 3212
 Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cgsex2g.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)
cgsex2g.2 (𝜒 → (𝜑𝜓))
Assertion
Ref Expression
cgsex2g ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem cgsex2g
StepHypRef Expression
1 cgsex2g.2 . . . 4 (𝜒 → (𝜑𝜓))
21biimpa 500 . . 3 ((𝜒𝜑) → 𝜓)
32exlimivv 1847 . 2 (∃𝑥𝑦(𝜒𝜑) → 𝜓)
4 elisset 3188 . . . . . 6 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 elisset 3188 . . . . . 6 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
64, 5anim12i 588 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7 eeanv 2170 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
86, 7sylibr 223 . . . 4 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
9 cgsex2g.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)
1092eximi 1753 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜒)
118, 10syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦𝜒)
121biimprcd 239 . . . . 5 (𝜓 → (𝜒𝜑))
1312ancld 574 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
14132eximdv 1835 . . 3 (𝜓 → (∃𝑥𝑦𝜒 → ∃𝑥𝑦(𝜒𝜑)))
1511, 14syl5com 31 . 2 ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦(𝜒𝜑)))
163, 15impbid2 215 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by: (None)
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