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Theorem elabreximdv 30267
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
elabreximdv.1 (𝐴 = 𝐵 → (𝜒𝜓))
elabreximdv.2 (𝜑𝐴𝑉)
elabreximdv.3 ((𝜑𝑥𝐶) → 𝜓)
Assertion
Ref Expression
elabreximdv ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabreximdv
StepHypRef Expression
1 nfv 1916 . 2 𝑥𝜑
2 nfv 1916 . 2 𝑥𝜒
3 elabreximdv.1 . 2 (𝐴 = 𝐵 → (𝜒𝜓))
4 elabreximdv.2 . 2 (𝜑𝐴𝑉)
5 elabreximdv.3 . 2 ((𝜑𝑥𝐶) → 𝜓)
61, 2, 3, 4, 5elabreximd 30266 1 ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wrex 3133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138
This theorem is referenced by: (None)
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