Theorem sbabel 2959

Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)

Hypothesis

Ref	Expression
sbabel.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
sbabel	⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐴(𝑥, 𝑦, 𝑧)

Proof of Theorem sbabel

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clabel 2910	. . . 4 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
2	1	sbbii 2113	. . 3 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
3		sbex 2318	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
4		sban 2117	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
5		sbabel.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2919	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
7	6	sbf 2308	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
8		sbv 2125	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
9	8	sbrbis 2346	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	9	sbalv 2208	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
11	7, 10	anbi12i 640	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
12	4, 11	bitri 278	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
13	12	exbii 1881	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14	2, 3, 13	3bitri 300	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
15		clabel 2910	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
16	14, 15	bitr4i 281	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  [wsb 2099   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914

This theorem is used by: (None)