Theorem sbcabel 3071

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 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sbcexg 3044	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵)))
3		sbcang 3033	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ∧ [𝐴 / 𝑥]𝑤 ∈ 𝐵)))
4		sbcalg 3042	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑)))
5		sbcbig 3036	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
6		sbcg 3059	. . . . . . . . . . . 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 1838	. . . . . . . . 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 2305	. . . . . . . . 9 ⊢ (𝑤 = {𝑦 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
12	11	sbcbii 3049	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
13		abeq2 2305	. . . . . . . 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 2333	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵
17	16	sbcgf 3057	. . . . . . 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 1839	. . . 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 2192	. . . 4 ⊢ ({𝑦 ∣ 𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
23	22	sbcbii 3049	. . 3 ⊢ ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
24		df-clel 2192	. . 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 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326  Vcvv 2763  [wsbc 2989

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  csbexga  4161