Theorem suceqd 6436

Description: Deduction associated with suceq 6437. (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 4595	. . 3 ⊢ (𝜑 → {𝐴} = {𝐵})
3	1, 2	uneq12d 4170	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6374	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6374	. 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 1569   ∪ cun 4151  {csn 4587  suc csuc 6370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-un 4153  df-sn 4588  df-suc 6374

This theorem is used by:  suceq  6437  scottrankd  9959  nosupbnd2  28085  bdayiun  28313  bdaypw2n0bndlem  28861  bdaypw2n0bnd  28862  z12bdaylem2  28869  rankscott  35817  fineqvnttrclselem3  35862  onprcf1acwevdlem2  36000