Theorem clelsb2 2857

Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2126). (Contributed by Jim Kingdon, 22-Nov-2018.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 24-Nov-2024.)

Assertion

Ref	Expression
clelsb2	⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2w 2813	. 2 ⊢ (𝑥 = 𝑧 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑧))
2		eleq2w 2813	. 2 ⊢ (𝑧 = 𝑦 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑦))
3	1, 2	sbievw2 2099	1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Syntax hints:   ↔ wb 206  [wsb 2065   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clel 2804

This theorem is referenced by: (None)