Theorem sbab 2881

Description: The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
sbab	⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴})

Distinct variable groups:   𝑧,𝐴   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem sbab

Step	Hyp	Ref	Expression
1		sbequ12 2242	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐴))
2	1	eqabdv 2866	1 ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴})

Syntax hints:   → wi 4   = wceq 1540  [wsb 2066   ∈ wcel 2105  {cab 2708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809

This theorem is referenced by:  sbcel12  4408  sbceqg  4409