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Theorem frege103 37079
Description: Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege103.z 𝑍𝑉
Assertion
Ref Expression
frege103 ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))

Proof of Theorem frege103
StepHypRef Expression
1 frege103.z . . 3 𝑍𝑉
21frege100 37076 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
3 frege19 36937 . 2 ((𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋)) → ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))))
42, 3ax-mp 5 1 ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1975  cun 3533   class class class wbr 4573   I cid 4934  cfv 5786  t+ctcl 13514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-frege1 36903  ax-frege2 36904  ax-frege8 36922  ax-frege52a 36970
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031
This theorem is referenced by:  frege104  37080
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