Theorem sbcel1v

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018) Avoid ax-13 . (Revised by Wolf Lammen, 30-Apr-2023)

	Ref	Expression
Assertion	sbcel1v	⊢ ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sbcex	⊢ ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 → 𝐴 ∈ V )
2		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
3		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ) )
4		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
5		clelsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 )
6	3 4 5	vtoclbg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
7	1 2 6	pm5.21nii	⊢ ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 )