Theorem suceqd 6456

Description: Deduction associated with suceq 6457. (Contributed by Rohan Ridenour, 8-Aug-2023.)

Hypothesis

Ref	Expression
suceqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
suceqd	⊢ (𝜑 → suc 𝐴 = suc 𝐵)

Proof of Theorem suceqd

Step	Hyp	Ref	Expression
1		suceqd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	sneqd 4596	. . 3 ⊢ (𝜑 → {𝐴} = {𝐵})
3	1, 2	uneq12d 4171	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6394	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6394	. 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
6	3, 4, 5	3eqtr4g 2823	1 ⊢ (𝜑 → suc 𝐴 = suc 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 4152  {csn 4588  suc csuc 6390

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-un 4154  df-sn 4589  df-suc 6394

This theorem is used by:  suceq  6457  ttrclss  9758  scottrankd  9986  nosupbnd2  28130  bdayiun  28358  bdaypw2n0bndlem  28906  bdaypw2n0bnd  28907  z12bdaylem2  28914  rankscott  35879  fineqvnttrclselem3  35932  onprcf1acwevdlem2  36087