Theorem wl-dfrabv 34897

Description: Alternate definition of restricted class abstraction (df-wl-rab 34895), when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 29-May-2023.)

Assertion

Ref	Expression
wl-dfrabv	⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrabv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-dfrabsb 34896	. 2 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}
2		clelsb3 2939	. . . . . 6 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
3		clelsb3 2939	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	2, 3	bitr4i 280	. . . . 5 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴)
5		sbco2vv 2107	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
6	4, 5	anbi12i 628	. . . 4 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
7		df-clab 2799	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ [𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
8		sban 2085	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
9	7, 8	bitri 277	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
10		df-clab 2799	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
11		sban 2085	. . . . 5 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
12	10, 11	bitri 277	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
13	6, 9, 12	3bitr4i 305	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
14	13	eqriv 2817	. 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
15	1, 14	eqtri 2843	1 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Syntax hints:   ∧ wa 398   = wceq 1536  [wsb 2068   ∈ wcel 2113  {cab 2798  {wl-crab 34870

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-8 2115  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-clel 2892  df-wl-rab 34895

This theorem is referenced by: (None)