Theorem suceqd 6432

Description: Deduction associated with suceq 6433. (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 4603	. . 3 ⊢ (𝜑 → {𝐴} = {𝐵})
3	1, 2	uneq12d 4123	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6370	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6370	. 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
6	3, 4, 5	3eqtr4g 2825	1 ⊢ (𝜑 → suc 𝐴 = suc 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3904  {csn 4591  suc csuc 6366

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-sn 4592  df-suc 6370

This theorem is used by:  suceq  6433  scottrankd  9885  nosupbnd2  27933  bdayiun  28161  bdaypw2n0bndlem  28709  bdaypw2n0bnd  28710  z12bdaylem2  28717  rankscott  35581  fineqvnttrclselem3  35595