Theorem wl-dfrabf 34311

Description: Alternate definition of restricted class abstraction (df-wl-rab 34307), when 𝑥 is not free in 𝐴. (Contributed by Wolf Lammen, 29-May-2023.)

Assertion

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

Proof of Theorem wl-dfrabf

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

Step	Hyp	Ref	Expression
1		wl-dfrabsb 34308	. 2 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}
2		nfnfc1 2935	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
3		id 22	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
4	2, 3	wl-clelsb3df 34310	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5		clelsb3 2893	. . . . . 6 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
6	4, 5	syl6rbbr 282	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
7		sbco2vv 2044	. . . . . 6 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
8	7	a1i 11	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	6, 8	anbi12d 621	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
10		df-clab 2759	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ [𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
11		sban 2031	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
12	10, 11	bitri 267	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
13		df-clab 2759	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
14		sban 2031	. . . . 5 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
15	13, 14	bitri 267	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
16	9, 12, 15	3bitr4g 306	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}))
17	16	eqrdv 2776	. 2 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
18	1, 17	syl5eq 2826	1 ⊢ (Ⅎ𝑥𝐴 → {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  [wsb 2015   ∈ wcel 2050  {cab 2758  Ⅎwnfc 2916  {wl-crab 34282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-wl-rab 34307

This theorem is referenced by: (None)