Theorem suceqd

Description: Deduction associated with suceq . (Contributed by Rohan Ridenour, 8-Aug-2023)

	Ref	Expression
Hypothesis	suceqd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	suceqd	⊢ ( 𝜑 → suc 𝐴 = suc 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		suceqd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2	1	sneqd	⊢ ( 𝜑 → { 𝐴 } = { 𝐵 } )
3	1 2	uneq12d	⊢ ( 𝜑 → ( 𝐴 ∪ { 𝐴 } ) = ( 𝐵 ∪ { 𝐵 } ) )
4		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
5		df-suc	⊢ suc 𝐵 = ( 𝐵 ∪ { 𝐵 } )
6	3 4 5	3eqtr4g	⊢ ( 𝜑 → suc 𝐴 = suc 𝐵 )