Theorem csbunig 3797

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 3106	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)})
2		sbcexg 3005	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
3		sbcang 2994	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵)))
4		sbcg 3020	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
5		sbcel2g 3066	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))
6	4, 5	anbi12d 465	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
7	3, 6	bitrd 187	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
8	7	exbidv 1813	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
9	2, 8	bitrd 187	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	abbidv 2284	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
11	1, 10	eqtrd 2198	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
12		df-uni 3790	. . 3 ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
13	12	csbeq2i 3072	. 2 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
14		df-uni 3790	. 2 ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
15	11, 13, 14	3eqtr4g 2224	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  [wsbc 2951  ⦋csb 3045  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046  df-uni 3790

This theorem is referenced by: (None)