Theorem wsuceq123 33808

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 1135	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝑅 = 𝑆)
2	1	cnveqd 5784	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆)
3		predeq123 6203	. . . 4 ⊢ ((◡𝑅 = ◡𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
4	2, 3	syld3an1 1409	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(◡𝑅, 𝐴, 𝑋) = Pred(◡𝑆, 𝐵, 𝑌))
5		simp2 1136	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵)
6	4, 5, 1	infeq123d 9240	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆))
7		df-wsuc 33806	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
8		df-wsuc 33806	. 2 ⊢ wsuc(𝑆, 𝐵, 𝑌) = inf(Pred(◡𝑆, 𝐵, 𝑌), 𝐵, 𝑆)
9	6, 7, 8	3eqtr4g 2803	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539  ^◡ccnv 5588  Predcpred 6201  infcinf 9200  wsuccwsuc 33804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-sup 9201  df-inf 9202  df-wsuc 33806

This theorem is referenced by:  wsuceq1  33809  wsuceq2  33810  wsuceq3  33811