Theorem sbabel 2339

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.)

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 1997	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
2		sban 1948	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ∧ [𝑦 / 𝑥]𝑣 ∈ 𝐴))
3		nfv 1521	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑣
4	3	sbf 1770	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
5	4	sbrbis 1954	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
6	5	sbalv 1998	. . . . . . 7 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
7		abeq2 2279	. . . . . . . 8 ⊢ (𝑣 = {𝑧 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
8	7	sbbii 1758	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
9		abeq2 2279	. . . . . . 7 ⊢ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	6, 8, 9	3bitr4i 211	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ↔ 𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑})
11		sbabel.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12	11	nfcri 2306	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
13	12	sbf 1770	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
14	10, 13	anbi12i 457	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ∧ [𝑦 / 𝑥]𝑣 ∈ 𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
15	2, 14	bitri 183	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
16	15	exbii 1598	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
17	1, 16	bitri 183	. 2 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
18		df-clel 2166	. . 3 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
19	18	sbbii 1758	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
20		df-clel 2166	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
21	17, 19, 20	3bitr4i 211	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485  [wsb 1755   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by: (None)