Theorem sbceqbid

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018)

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

Proof

Step	Hyp	Ref	Expression
1		sbceqbid.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sbceqbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	2	abbidv	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜒 } )
4	1 3	eleq12d	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝐵 ∈ { 𝑥 ∣ 𝜒 } ) )
5		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
6		df-sbc	⊢ ( [ 𝐵 / 𝑥 ] 𝜒 ↔ 𝐵 ∈ { 𝑥 ∣ 𝜒 } )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜒 ) )