Theorem sbabel 2977

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 2623	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
2		sban 2558	. . . . 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 2942	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
5	4	sbf 2539	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
6		nfv 2005	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑣
7	6	sbf 2539	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
8	7	sbrbis 2564	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
9	8	sbalv 2624	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	5, 9	anbi12i 614	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
11	2, 10	bitri 266	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
12	11	exbii 1933	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
13	1, 12	bitri 266	. 2 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14		clabel 2933	. . 3 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
15	14	sbbii 2067	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
16		clabel 2933	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
17	13, 15, 16	3bitr4i 294	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Syntax hints:   ↔ wb 197   ∧ wa 384  ∀wal 1635  ∃wex 1859  [wsb 2060   ∈ wcel 2156  {cab 2792  Ⅎwnfc 2935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937

This theorem is referenced by: (None)