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Theorem cgsexg 3269
 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 500 . . 3 ((𝜒𝜑) → 𝜓)
32exlimiv 1898 . 2 (∃𝑥(𝜒𝜑) → 𝜓)
4 elisset 3246 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 cgsexg.1 . . . . 5 (𝑥 = 𝐴𝜒)
65eximi 1802 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
74, 6syl 17 . . 3 (𝐴𝑉 → ∃𝑥𝜒)
81biimprcd 240 . . . . 5 (𝜓 → (𝜒𝜑))
98ancld 575 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
109eximdv 1886 . . 3 (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒𝜑)))
117, 10syl5com 31 . 2 (𝐴𝑉 → (𝜓 → ∃𝑥(𝜒𝜑)))
123, 11impbid2 216 1 (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233 This theorem is referenced by: (None)
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