Theorem sbabel 2266

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 1940	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
2		sban 1889	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ∧ [𝑦 / 𝑥]𝑣 ∈ 𝐴))
3		nfv 1476	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑣
4	3	sbf 1718	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
5	4	sbrbis 1895	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
6	5	sbalv 1941	. . . . . . 7 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
7		abeq2 2208	. . . . . . . 8 ⊢ (𝑣 = {𝑧 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
8	7	sbbii 1706	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
9		abeq2 2208	. . . . . . 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 2234	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
13	12	sbf 1718	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
14	10, 13	anbi12i 451	. . . . 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 1552	. . 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 2096	. . 3 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
19	18	sbbii 1706	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
20		df-clel 2096	. 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 1297   = wceq 1299  ∃wex 1436   ∈ wcel 1448  [wsb 1703  {cab 2086  Ⅎwnfc 2227

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229

This theorem is referenced by: (None)