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Theorem cgsex2g 2607
 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 284 . . 3 ((𝜒𝜑) → 𝜓)
32exlimivv 1792 . 2 (∃𝑥𝑦(𝜒𝜑) → 𝜓)
4 elisset 2585 . . . . . 6 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 elisset 2585 . . . . . 6 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
64, 5anim12i 325 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7 eeanv 1823 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
86, 7sylibr 141 . . . 4 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
9 cgsex2g.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)
1092eximi 1508 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜒)
118, 10syl 14 . . 3 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦𝜒)
121biimprcd 153 . . . . 5 (𝜓 → (𝜒𝜑))
1312ancld 312 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
14132eximdv 1778 . . 3 (𝜓 → (∃𝑥𝑦𝜒 → ∃𝑥𝑦(𝜒𝜑)))
1511, 14syl5com 29 . 2 ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦(𝜒𝜑)))
163, 15impbid2 135 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259  ∃wex 1397   ∈ wcel 1409 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576 This theorem is referenced by: (None)
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