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Theorem cgsexg 2772
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 296 . . 3 ((𝜒𝜑) → 𝜓)
32exlimiv 1598 . 2 (∃𝑥(𝜒𝜑) → 𝜓)
4 elisset 2751 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 cgsexg.1 . . . . 5 (𝑥 = 𝐴𝜒)
65eximi 1600 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
74, 6syl 14 . . 3 (𝐴𝑉 → ∃𝑥𝜒)
81biimprcd 160 . . . . 5 (𝜓 → (𝜒𝜑))
98ancld 325 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
109eximdv 1880 . . 3 (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒𝜑)))
117, 10syl5com 29 . 2 (𝐴𝑉 → (𝜓 → ∃𝑥(𝜒𝜑)))
123, 11impbid2 143 1 (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by: (None)
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