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Theorem elabreximdv 32538
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 1911 . 2 𝑥𝜑
2 nfv 1911 . 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 32537 1 ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068
This theorem is referenced by: (None)
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