Theorem wl-dfrabsb 34897

Description: Alternate definition of restricted class abstraction (df-wl-rab 34896), 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 34896	. 2 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))}
2		sb6 2092	. . . 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 2885	. 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))}
5	1, 4	eqtr4i 2846	1 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534   = wceq 1536  [wsb 2068   ∈ wcel 2113  {cab 2798  {wl-crab 34871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-wl-rab 34896

This theorem is referenced by:  wl-dfrabv  34898  wl-dfrabf  34900