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Theorem frege102d 38887
 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 39099. (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 474 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3 frege102d.a . . . 4 (𝜑𝐴 ∈ V)
43adantr 474 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5 frege102d.b . . . 4 (𝜑𝐵 ∈ V)
65adantr 474 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7 frege102d.c . . . 4 (𝜑𝐶 ∈ V)
87adantr 474 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9 simpr 479 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10 frege102d.cb . . . 4 (𝜑𝐶𝑅𝐵)
1110adantr 474 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
122, 4, 6, 8, 9, 11frege96d 38882 . 2 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
131adantr 474 . . 3 ((𝜑𝐴 = 𝐶) → 𝑅 ∈ V)
14 simpr 479 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1510adantr 474 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐶𝑅𝐵)
1614, 15eqbrtrd 4895 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴𝑅𝐵)
1713, 16frege91d 38884 . 2 ((𝜑𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18 frege102d.ac . 2 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
1912, 17, 18mpjaodan 988 1 (𝜑𝐴(t+‘𝑅)𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166  Vcvv 3414   class class class wbr 4873  ‘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-8 2168  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-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  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-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fv 6131  df-trcl 14105 This theorem is referenced by:  frege108d  38889
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