Theorem sbabel 2940

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 2884	. . . 4 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
2	1	sbbii 2080	. . 3 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
3		sbex 2281	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)))
4		sban 2084	. . . . 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 2893	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
7	6	sbf 2266	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
8		sbv 2092	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
9	8	sbrbis 2310	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	9	sbalv 2162	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
11	7, 10	anbi12i 626	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 ∈ 𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
12	4, 11	bitri 274	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
13	12	exbii 1851	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑)) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14	2, 3, 13	3bitri 296	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
15		clabel 2884	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑)))
16	14, 15	bitr4i 277	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  [wsb 2068   ∈ wcel 2108  {cab 2715  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888

This theorem is referenced by: (None)