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Theorem frege107 39104
 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 39103 . 2 (𝑍(t+‘𝑅)𝑉𝑍((t+‘𝑅) ∪ I )𝑉)
3 frege7 38942 . 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 2166   ∪ cun 3796   class class class wbr 4873   I cid 5249  ‘cfv 6123  t+ctcl 14103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege28 38964  ax-frege31 38968  ax-frege52a 38991 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349 This theorem is referenced by:  frege108  39105
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