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Theorem frege102d 39977
Description: If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 40189. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege102d.r (𝜑𝑅 ∈ V)
frege102d.a (𝜑𝐴 ∈ V)
frege102d.b (𝜑𝐵 ∈ V)
frege102d.c (𝜑𝐶 ∈ V)
frege102d.ac (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
frege102d.cb (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
frege102d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege102d
StepHypRef Expression
1 frege102d.r . . . 4 (𝜑𝑅 ∈ V)
21adantr 481 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3 frege102d.a . . . 4 (𝜑𝐴 ∈ V)
43adantr 481 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5 frege102d.b . . . 4 (𝜑𝐵 ∈ V)
65adantr 481 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7 frege102d.c . . . 4 (𝜑𝐶 ∈ V)
87adantr 481 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9 simpr 485 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10 frege102d.cb . . . 4 (𝜑𝐶𝑅𝐵)
1110adantr 481 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
122, 4, 6, 8, 9, 11frege96d 39972 . 2 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
131adantr 481 . . 3 ((𝜑𝐴 = 𝐶) → 𝑅 ∈ V)
14 simpr 485 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1510adantr 481 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐶𝑅𝐵)
1614, 15eqbrtrd 5079 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴𝑅𝐵)
1713, 16frege91d 39974 . 2 ((𝜑𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18 frege102d.ac . 2 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
1912, 17, 18mpjaodan 952 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  Vcvv 3492   class class class wbr 5057  cfv 6348  t+ctcl 14333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-trcl 14335
This theorem is referenced by:  frege108d  39979
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