Theorem sbceq1d

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

	Ref	Expression
Hypothesis	sbceq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	sbceq1d	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )

Proof

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