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Theorem cgsexg 2721
 Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
Hypotheses
Ref Expression
cgsexg.1 (𝑥 = 𝐴𝜒)
cgsexg.2 (𝜒 → (𝜑𝜓))
Assertion
Ref Expression
cgsexg (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝑉(𝑥)

Proof of Theorem cgsexg
StepHypRef Expression
1 cgsexg.2 . . . 4 (𝜒 → (𝜑𝜓))
21biimpa 294 . . 3 ((𝜒𝜑) → 𝜓)
32exlimiv 1577 . 2 (∃𝑥(𝜒𝜑) → 𝜓)
4 elisset 2700 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 cgsexg.1 . . . . 5 (𝑥 = 𝐴𝜒)
65eximi 1579 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
74, 6syl 14 . . 3 (𝐴𝑉 → ∃𝑥𝜒)
81biimprcd 159 . . . . 5 (𝜓 → (𝜒𝜑))
98ancld 323 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
109eximdv 1852 . . 3 (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒𝜑)))
117, 10syl5com 29 . 2 (𝐴𝑉 → (𝜓 → ∃𝑥(𝜒𝜑)))
123, 11impbid2 142 1 (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688 This theorem is referenced by: (None)
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