Theorem sbcabel 3861

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 3513	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sbcex2 3834	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
3		sbcan 3821	. . . . . 6 ⊢ ([𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ∧ [𝐴 / 𝑥]𝑤 ∈ 𝐵))
4		sbcal 3833	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑))
5		sbcbig 3823	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
6		sbcg 3847	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
7	6	bibi1d 346	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
8	5, 7	bitrd 281	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ (𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
9	8	albidv 1917	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
10	4, 9	syl5bb 285	. . . . . . . 8 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑)))
11		abeq2 2945	. . . . . . . . 9 ⊢ (𝑤 = {𝑦 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
12	11	sbcbii 3829	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝑤 ↔ 𝜑))
13		abeq2 2945	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ [𝐴 / 𝑥]𝜑))
14	10, 12, 13	3bitr4g 316	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ↔ 𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
15		sbcabel.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
16	15	nfcri 2971	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵
17	16	sbcgf 3845	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
18	14, 17	anbi12d 632	. . . . . 6 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑤 = {𝑦 ∣ 𝜑} ∧ [𝐴 / 𝑥]𝑤 ∈ 𝐵) ↔ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
19	3, 18	syl5bb 285	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ (𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
20	19	exbidv 1918	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
21	2, 20	syl5bb 285	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵)))
22		dfclel 2894	. . . 4 ⊢ ({𝑦 ∣ 𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
23	22	sbcbii 3829	. . 3 ⊢ ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤(𝑤 = {𝑦 ∣ 𝜑} ∧ 𝑤 ∈ 𝐵))
24		dfclel 2894	. . 3 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∧ 𝑤 ∈ 𝐵))
25	21, 23, 24	3bitr4g 316	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))
26	1, 25	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  Ⅎwnfc 2961  Vcvv 3495  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-sbc 3773

This theorem is referenced by:  csbexg  5207