Theorem sbceq1dd

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017) (Revised by NM, 30-Jun-2018)

	Ref	Expression
Hypotheses	sbceq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sbceq1dd.2	⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )
Assertion	sbceq1dd	⊢ ( 𝜑 → [ 𝐵 / 𝑥 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbceq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sbceq1dd.2	⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )
3	1	sbceq1d	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )
4	2 3	mpbid	⊢ ( 𝜑 → [ 𝐵 / 𝑥 ] 𝜓 )