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Theorem wsuceq123 35795
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 1135 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
21cnveqd 5888 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
3 predeq123 6323 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
42, 3syld3an1 1409 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
5 simp2 1136 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
64, 5, 1infeq123d 9518 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7 df-wsuc 35793 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8 df-wsuc 35793 . 2 wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
96, 7, 83eqtr4g 2799 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  ccnv 5687  Predcpred 6321  infcinf 9478  wsuccwsuc 35791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-sup 9479  df-inf 9480  df-wsuc 35793
This theorem is referenced by:  wsuceq1  35796  wsuceq2  35797  wsuceq3  35798
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