Theorem clelsb1f

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Usage of this theorem is discouraged because it depends on ax-13 . See clelsb1fw not requiring ax-13 , but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Revised by Thierry Arnoux, 13-Mar-2017) (Proof shortened by Wolf Lammen, 7-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypothesis	clelsb1f.1	⊢ Ⅎ 𝑥 𝐴
Assertion	clelsb1f	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		clelsb1f.1	⊢ Ⅎ 𝑥 𝐴
2	1	nfcri	⊢ Ⅎ 𝑥 𝑤 ∈ 𝐴
3	2	sbco2	⊢ ( [ 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 )