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Theorem cgsexg 2606
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 284 . . 3 ((𝜒𝜑) → 𝜓)
32exlimiv 1505 . 2 (∃𝑥(𝜒𝜑) → 𝜓)
4 elisset 2585 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 cgsexg.1 . . . . 5 (𝑥 = 𝐴𝜒)
65eximi 1507 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
74, 6syl 14 . . 3 (𝐴𝑉 → ∃𝑥𝜒)
81biimprcd 153 . . . . 5 (𝜓 → (𝜒𝜑))
98ancld 312 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
109eximdv 1776 . . 3 (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒𝜑)))
117, 10syl5com 29 . 2 (𝐴𝑉 → (𝜓 → ∃𝑥(𝜒𝜑)))
123, 11impbid2 135 1 (𝐴𝑉 → (∃𝑥(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-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-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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