Theorem clelsb1fw

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Version of clelsb1f with a disjoint variable condition, which does not require ax-13 . (Contributed by Rodolfo Medina, 28-Apr-2010) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypothesis	clelsb1fw.1	⊢ Ⅎ 𝑥 𝐴
Assertion	clelsb1fw	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		clelsb1fw.1	⊢ Ⅎ 𝑥 𝐴
2	1	nfcri	⊢ Ⅎ 𝑥 𝑤 ∈ 𝐴
3	2	sbco2v	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑤 ] 𝑤 ∈ 𝐴 ↔ [ 𝑦 / 𝑤 ] 𝑤 ∈ 𝐴 )
4		clelsb1	⊢ ( [ 𝑥 / 𝑤 ] 𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 )
5	4	sbbii	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑤 ] 𝑤 ∈ 𝐴 ↔ [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 )
6		clelsb1	⊢ ( [ 𝑦 / 𝑤 ] 𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )
7	3 5 6	3bitr3i	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )