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Theorem frege101 40596
Description: Lemma for frege102 40597. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege99.z 𝑍𝑈
Assertion
Ref Expression
frege101 ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))))

Proof of Theorem frege101
StepHypRef Expression
1 frege99.z . . 3 𝑍𝑈
21frege100 40595 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
3 frege48 40484 . 2 ((𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋)) → ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)))))
42, 3ax-mp 5 1 ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2114  cun 3906   class class class wbr 5042   I cid 5436  cfv 6334  t+ctcl 14336
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-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-frege1 40422  ax-frege2 40423  ax-frege8 40441  ax-frege28 40462  ax-frege31 40466  ax-frege41 40477  ax-frege52a 40489
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539
This theorem is referenced by:  frege102  40597
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