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Theorem wsuceq123 33284
 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 5714 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
3 predeq123 6124 . . . 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 8947 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7 df-wsuc 33282 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8 df-wsuc 33282 . 2 wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
96, 7, 83eqtr4g 2858 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 1538  ◡ccnv 5522  Predcpred 6122  infcinf 8907  wsuccwsuc 33280 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-sup 8908  df-inf 8909  df-wsuc 33282 This theorem is referenced by:  wsuceq1  33285  wsuceq2  33286  wsuceq3  33287
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