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Theorem frege107 43934
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 43933 . 2 (𝑍(t+‘𝑅)𝑉𝑍((t+‘𝑅) ∪ I )𝑉)
3 frege7 43772 . 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 2108  cun 3974   class class class wbr 5166   I cid 5592  cfv 6575  t+ctcl 15036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege1 43754  ax-frege2 43755  ax-frege8 43773  ax-frege28 43794  ax-frege31 43798  ax-frege52a 43821
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707
This theorem is referenced by:  frege108  43935
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