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Theorem cgsex2g 2655
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 290 . . 3 ((𝜒𝜑) → 𝜓)
32exlimivv 1824 . 2 (∃𝑥𝑦(𝜒𝜑) → 𝜓)
4 elisset 2633 . . . . . 6 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 elisset 2633 . . . . . 6 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
64, 5anim12i 331 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7 eeanv 1855 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
86, 7sylibr 132 . . . 4 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
9 cgsex2g.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)
1092eximi 1537 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜒)
118, 10syl 14 . . 3 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦𝜒)
121biimprcd 158 . . . . 5 (𝜓 → (𝜒𝜑))
1312ancld 318 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
14132eximdv 1810 . . 3 (𝜓 → (∃𝑥𝑦𝜒 → ∃𝑥𝑦(𝜒𝜑)))
1511, 14syl5com 29 . 2 ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦(𝜒𝜑)))
163, 15impbid2 141 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by: (None)
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