Theorem elsb2 2123

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2114 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)

Assertion

Ref	Expression
elsb2	⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2121	. 2 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
2		elequ2 2121	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
3	1, 2	sbievw2 2099	1 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 205  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068

This theorem is referenced by:  nfnid  5298