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Theorem frege107 43279
Description: Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege107.v 𝑉𝐴
Assertion
Ref Expression
frege107 ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉)))

Proof of Theorem frege107
StepHypRef Expression
1 frege107.v . . 3 𝑉𝐴
21frege106 43278 . 2 (𝑍(t+‘𝑅)𝑉𝑍((t+‘𝑅) ∪ I )𝑉)
3 frege7 43117 . 2 ((𝑍(t+‘𝑅)𝑉𝑍((t+‘𝑅) ∪ I )𝑉) → ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉))))
42, 3ax-mp 5 1 ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cun 3941   class class class wbr 5141   I cid 5566  cfv 6536  t+ctcl 14935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-frege1 43099  ax-frege2 43100  ax-frege8 43118  ax-frege28 43139  ax-frege31 43143  ax-frege52a 43166
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676
This theorem is referenced by:  frege108  43280
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