Theorem suceqd 6392

Description: Deduction associated with suceq 6393. (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 4594	. . 3 ⊢ (𝜑 → {𝐴} = {𝐵})
3	1, 2	uneq12d 4123	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4		df-suc 6331	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		df-suc 6331	. 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
6	3, 4, 5	3eqtr4g 2797	1 ⊢ (𝜑 → suc 𝐴 = suc 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3901  {csn 4582  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-suc 6331

This theorem is referenced by:  suceq  6393  nosupbnd2  27699  bdayiun  27926  bdaypw2n0bndlem  28474  bdaypw2n0bnd  28475  z12bdaylem2  28482  fineqvnttrclselem3  35305  scottrankd  44608