Theorem csbunig 3896

Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbunig	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbunig

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabg 3186	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)})
2		sbcexg 3083	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
3		sbcang 3072	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵)))
4		sbcg 3098	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
5		sbcel2g 3145	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))
6	4, 5	anbi12d 473	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
7	3, 6	bitrd 188	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
8	7	exbidv 1871	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
9	2, 8	bitrd 188	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	abbidv 2347	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
11	1, 10	eqtrd 2262	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
12		df-uni 3889	. . 3 ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
13	12	csbeq2i 3151	. 2 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
14		df-uni 3889	. 2 ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
15	11, 13, 14	3eqtr4g 2287	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  [wsbc 3028  ⦋csb 3124  ∪ cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125  df-uni 3889

This theorem is referenced by: (None)