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Theorem wsuceq123 35319
Description: Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
Assertion
Ref Expression
wsuceq123 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Proof of Theorem wsuceq123
StepHypRef Expression
1 simp1 1133 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
21cnveqd 5869 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
3 predeq123 6295 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
42, 3syld3an1 1407 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
5 simp2 1134 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
64, 5, 1infeq123d 9478 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7 df-wsuc 35317 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8 df-wsuc 35317 . 2 wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
96, 7, 83eqtr4g 2791 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  ccnv 5668  Predcpred 6293  infcinf 9438  wsuccwsuc 35315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-sup 9439  df-inf 9440  df-wsuc 35317
This theorem is referenced by:  wsuceq1  35320  wsuceq2  35321  wsuceq3  35322
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