Theorem sbabel 2966

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, 26-Dec-2019.)

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		sbex 2501	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
2		sban 2031	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
3		sbabel.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2926	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
5	4	sbf 2199	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
6		nfv 1873	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑣
7	6	sbf 2199	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
8	7	sbrbis 2243	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
9	8	sbalv 2502	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	5, 9	anbi12i 617	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
11	2, 10	bitri 267	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
12	11	exbii 1810	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
13	1, 12	bitri 267	. 2 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14		clabel 2914	. . 3 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
15	14	sbbii 2027	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
16		clabel 2914	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
17	13, 15, 16	3bitr4i 295	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Syntax hints:   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742  [wsb 2015   ∈ wcel 2050  {cab 2758  Ⅎwnfc 2916

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-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

This theorem is referenced by: (None)