Theorem wl-dfrabf 34900

Description: Alternate definition of restricted class abstraction (df-wl-rab 34896), 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 34897	. 2 ⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}
2		nfnfc1 2979	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
3		id 22	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
4	2, 3	wl-clelsb3df 34899	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5		clelsb3 2939	. . . . . 6 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
6	4, 5	syl6rbbr 292	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
7		sbco2vv 2107	. . . . . 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 632	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
10		df-clab 2799	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ [𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
11		sban 2085	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
12	10, 11	bitri 277	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
13		df-clab 2799	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
14		sban 2085	. . . . 5 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
15	13, 14	bitri 277	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
16	9, 12, 15	3bitr4g 316	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}))
17	16	eqrdv 2818	. 2 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
18	1, 17	syl5eq 2867	1 ⊢ (Ⅎ𝑥𝐴 → {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536  [wsb 2068   ∈ wcel 2113  {cab 2798  Ⅎwnfc 2960  {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-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-wl-rab 34896

This theorem is referenced by: (None)