Theorem wsuceq1 36379

Description: Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
wsuceq1	⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))

Proof of Theorem wsuceq1

Step	Hyp	Ref	Expression
1		eqid 2762	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2762	. 2 ⊢ 𝑋 = 𝑋
3		wsuceq123 36378	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑋) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))
4	1, 2, 3	mp3an23 1482	1 ⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  wsuccwsuc 36374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-sup 9415  df-inf 9416  df-wsuc 36376

This theorem is used by: (None)