Theorem suceqd 6430

Description: Deduction associated with suceq 6431. (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 4116	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6368	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6368	. 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
6	3, 4, 5	3eqtr4g 2821	1 ⊢ (𝜑 → suc 𝐴 = suc 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3897  {csn 4584  suc csuc 6364

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-sn 4585  df-suc 6368

This theorem is used by:  suceq  6431  scottrankd  9949  nosupbnd2  28073  bdayiun  28301  bdaypw2n0bndlem  28849  bdaypw2n0bnd  28850  z12bdaylem2  28857  rankscott  35752  fineqvnttrclselem3  35791  onprcf1acwevdlem2  35896