Theorem clelsb2 2888

Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2161). (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 2844	. 2 ⊢ (𝑥 = 𝑧 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑧))
2		eleq2w 2844	. 2 ⊢ (𝑧 = 𝑦 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑦))
3	1, 2	sbievw2 2134	1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2098   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clel 2835

This theorem is used by: (None)