Theorem elsb3 2119

Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)

Assertion

Ref	Expression
elsb3	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)

Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2118	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
2		elequ1 2118	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
3	1, 2	sbievw2 2104	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)

Syntax hints:   ↔ wb 209  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070

This theorem is referenced by:  cvjust  2793