Theorem wsuceq123 35809

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 1136	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝑅 = 𝑆)
2	1	cnveqd 5842	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
3		predeq123 6278	. . . 4 ⊢ ((◡𝑅 = ◡𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
4	2, 3	syld3an1 1412	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
5		simp2 1137	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵)
6	4, 5, 1	infeq123d 9440	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7		df-wsuc 35807	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8		df-wsuc 35807	. 2 ⊢ wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
9	6, 7, 8	3eqtr4g 2790	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540  ^◡ccnv 5640  Predcpred 6276  infcinf 9399  wsuccwsuc 35805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-sup 9400  df-inf 9401  df-wsuc 35807

This theorem is referenced by:  wsuceq1  35810  wsuceq2  35811  wsuceq3  35812