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Theorem cgsex2g 3513
 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 480 . . 3 ((𝜒𝜑) → 𝜓)
32exlimivv 1933 . 2 (∃𝑥𝑦(𝜒𝜑) → 𝜓)
4 elisset 3480 . . . . . 6 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 elisset 3480 . . . . . 6 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
64, 5anim12i 615 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7 exdistrv 1956 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
86, 7sylibr 237 . . . 4 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
9 cgsex2g.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)
1092eximi 1837 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜒)
118, 10syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦𝜒)
121biimprcd 253 . . . . 5 (𝜓 → (𝜒𝜑))
1312ancld 554 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
14132eximdv 1920 . . 3 (𝜓 → (∃𝑥𝑦𝜒 → ∃𝑥𝑦(𝜒𝜑)))
1511, 14syl5com 31 . 2 ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦(𝜒𝜑)))
163, 15impbid2 229 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2815  df-clel 2894 This theorem is referenced by: (None)
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