Theorem sbcabel 3124

Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Hypothesis

Ref	Expression
sbcabel.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
sbcabel	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcabel

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sbcexg 3096	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵)))
3		sbcang 3085	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ∧ [𝐴 / 𝑥]𝑤 ∈ 𝐵)))
4		sbcalg 3094	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑)))
5		sbcbig 3088	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
6		sbcg 3111	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
7	6	bibi1d 233	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
8	5, 7	bitrd 188	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ (𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
9	8	albidv 1873	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
10	4, 9	bitrd 188	. . . . . . . 8 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
11		abeq2 2341	. . . . . . . . 9 ⊢ (𝑤 = {𝑦 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
12	11	sbcbii 3101	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
13		abeq2 2341	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑))
14	10, 12, 13	3bitr4g 223	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ↔ 𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
15		sbcabel.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
16	15	nfcri 2378	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵
17	16	sbcgf 3109	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
18	14, 17	anbi12d 473	. . . . . 6 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ∧ [𝐴 / 𝑥]𝑤 ∈ 𝐵) ↔ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
19	3, 18	bitrd 188	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
20	19	exbidv 1874	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
21	2, 20	bitrd 188	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
22		df-clel 2228	. . . 4 ⊢ ({𝑦 ∣ 𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
23	22	sbcbii 3101	. . 3 ⊢ ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
24		df-clel 2228	. . 3 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵))
25	21, 23, 24	3bitr4g 223	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))
26	1, 25	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {cab 2218  Ⅎwnfc 2371  Vcvv 2812  [wsbc 3041

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sbc 3042

This theorem is referenced by:  csbexga  4237