Theorem wsuceq123 31760

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 1061	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝑅 = 𝑆)
2	1	cnveqd 5298	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
3		predeq123 5681	. . . 4 ⊢ ((◡𝑅 = ◡𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
4	2, 3	syld3an1 1372	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
5		simp2 1062	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵)
6	4, 5, 1	infeq123d 8387	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7		df-wsuc 31756	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8		df-wsuc 31756	. 2 ⊢ wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
9	6, 7, 8	3eqtr4g 2681	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483  ^◡ccnv 5113  Predcpred 5679  infcinf 8347  wsuccwsuc 31752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-sup 8348  df-inf 8349  df-wsuc 31756

This theorem is referenced by:  wsuceq1  31761  wsuceq2  31762  wsuceq3  31763