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Theorem srcmpltd 32420
Description: If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
Hypotheses
Ref Expression
srcmpltd.1 (𝜑 → (𝐶𝐴𝜓))
srcmpltd.2 (𝜑 → (𝐶 ∈ (𝐵𝐴) → 𝜓))
Assertion
Ref Expression
srcmpltd (𝜑 → (𝐶𝐵𝜓))

Proof of Theorem srcmpltd
StepHypRef Expression
1 elun2 4128 . . 3 (𝐶𝐵𝐶 ∈ (𝐴𝐵))
2 undif2 4397 . . 3 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
31, 2eleqtrrdi 2925 . 2 (𝐶𝐵𝐶 ∈ (𝐴 ∪ (𝐵𝐴)))
4 srcmpltd.1 . . 3 (𝜑 → (𝐶𝐴𝜓))
5 srcmpltd.2 . . 3 (𝜑 → (𝐶 ∈ (𝐵𝐴) → 𝜓))
6 elunant 4129 . . 3 ((𝐶 ∈ (𝐴 ∪ (𝐵𝐴)) → 𝜓) ↔ ((𝐶𝐴𝜓) ∧ (𝐶 ∈ (𝐵𝐴) → 𝜓)))
74, 5, 6sylanbrc 586 . 2 (𝜑 → (𝐶 ∈ (𝐴 ∪ (𝐵𝐴)) → 𝜓))
83, 7syl5 34 1 (𝜑 → (𝐶𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  cdif 3905  cun 3906
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794
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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266
This theorem is referenced by:  prsrcmpltd  32421
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