Theorem wsuceq123 36014

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 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Proof of Theorem wsuceq123

Step	Hyp	Ref	Expression
1		simp1 1137	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝑅 = 𝑆)
2	1	cnveqd 5826	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
3		predeq123 6262	. . . 4 ⊢ ((◡𝑅 = ◡𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
4	2, 3	syld3an1 1413	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
5		simp2 1138	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵)
6	4, 5, 1	infeq123d 9390	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7		df-wsuc 36012	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8		df-wsuc 36012	. 2 ⊢ wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
9	6, 7, 8	3eqtr4g 2797	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542  ^◡ccnv 5625  Predcpred 6260  infcinf 9349  wsuccwsuc 36010

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5632  df-cnv 5634  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-sup 9350  df-inf 9351  df-wsuc 36012

This theorem is referenced by:  wsuceq1  36015  wsuceq2  36016  wsuceq3  36017