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Theorem frege102d 42264
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 42475. (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 42259 . 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 5162 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴𝑅𝐵)
1713, 16frege91d 42261 . 2 ((𝜑𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18 frege102d.ac . 2 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
1912, 17, 18mpjaodan 957 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3472   class class class wbr 5140  cfv 6531  t+ctcl 14913
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pow 5355  ax-pr 5419  ax-un 7707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4943  df-br 5141  df-opab 5203  df-mpt 5224  df-id 5566  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-iota 6483  df-fun 6533  df-fv 6539  df-trcl 14915
This theorem is referenced by:  frege108d  42266
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