Theorem sbab 2906

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 2286	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐴))
2	1	eqabdv 2893	1 ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴})

This proof depends on syntax axioms:   → wi 4   = wceq 1569  [wsb 2098   ∈ wcel 2144  {cab 2738

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  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835

This theorem is used by:  sbcel12  4368  sbceqg  4369