Theorem sbabel 2419

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 2064	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
2		sban 2015	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ∧ [𝑦 / 𝑥]𝑣 ∈ 𝐴))
3		nfv 1581	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑣
4	3	sbf 1830	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣)
5	4	sbrbis 2021	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝑣 ↔ 𝜑) ↔ (𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
6	5	sbalv 2065	. . . . . . 7 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
7		abeq2 2347	. . . . . . . 8 ⊢ (𝑣 = {𝑧 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
8	7	sbbii 1818	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝑣 ↔ 𝜑))
9		abeq2 2347	. . . . . . 7 ⊢ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑣 ↔ [𝑦 / 𝑥]𝜑))
10	6, 8, 9	3bitr4i 212	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ↔ 𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑})
11		sbabel.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12	11	nfcri 2386	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐴
13	12	sbf 1830	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴)
14	10, 13	anbi12i 464	. . . . 5 ⊢ (([𝑦 / 𝑥]𝑣 = {𝑧 ∣ 𝜑} ∧ [𝑦 / 𝑥]𝑣 ∈ 𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
15	2, 14	bitri 184	. . . 4 ⊢ ([𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
16	15	exbii 1658	. . 3 ⊢ (∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
17	1, 16	bitri 184	. 2 ⊢ ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
18		df-clel 2234	. . 3 ⊢ ({𝑧 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
19	18	sbbii 1818	. 2 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧 ∣ 𝜑} ∧ 𝑣 ∈ 𝐴))
20		df-clel 2234	. 2 ⊢ ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣 ∈ 𝐴))
21	17, 19, 20	3bitr4i 212	1 ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545  [wsb 1815   ∈ wcel 2209  {cab 2224  Ⅎwnfc 2379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381

This theorem is referenced by: (None)