Theorem suceqd 6429

Description: Deduction associated with suceq 6430. (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 4606	. . 3 ⊢ (𝜑 → {𝐴} = {𝐵})
3	1, 2	uneq12d 4131	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6367	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6367	. 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
6	3, 4, 5	3eqtr4g 2829	1 ⊢ (𝜑 → suc 𝐴 = suc 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∪ cun 3911  {csn 4594  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595  df-suc 6367

This theorem is referenced by:  suceq  6430  scottrankd  9874  nosupbnd2  27846  bdayiun  28074  bdaypw2n0bndlem  28622  bdaypw2n0bnd  28623  z12bdaylem2  28630  fineqvnttrclselem3  35459