Theorem wl-dfrabsb 34876

Description: Alternate definition of restricted class abstraction (df-wl-rab 34875), using substitution. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

Ref	Expression
wl-dfrabsb	⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}

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

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfrabsb

Step	Hyp	Ref	Expression
1		df-wl-rab 34875	. 2 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))}
2		sb6 2093	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	anbi2i 624	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	abbii 2886	. 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))}
5	1, 4	eqtr4i 2847	1 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537  [wsb 2069   ∈ wcel 2114  {cab 2799  {wl-crab 34850

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 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-wl-rab 34875

This theorem is referenced by:  wl-dfrabv  34877  wl-dfrabf  34879