Theorem sbcrext 3038

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcrext	⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵

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

Proof of Theorem sbcrext

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2969	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
2	1	a1i 9	. 2 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V))
3		nfnfc1 2320	. . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
4		id 19	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
5		nfcvd 2318	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦V)
6	4, 5	nfeld 2333	. . 3 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7		sbcex 2969	. . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
8	7	2a1i 27	. . 3 ⊢ (Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)))
9	3, 6, 8	rexlimd2 2590	. 2 ⊢ (Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))
10		sbcco 2982	. . . 4 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
11		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → 𝐴 ∈ V)
12		sbsbc 2964	. . . . . . 7 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
13		nfcv 2317	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
14		nfs1v 1937	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
15	13, 14	nfrexxy 2514	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
16		sbequ12 1769	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
17	16	rexbidv 2476	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
18	15, 17	sbie 1789	. . . . . . 7 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
19	12, 18	bitr3i 186	. . . . . 6 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
20		nfcvd 2318	. . . . . . . . . 10 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝑧)
21	20, 4	nfeqd 2332	. . . . . . . . 9 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
22	3, 21	nfan1 1562	. . . . . . . 8 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴)
23		dfsbcq2 2963	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
24	23	adantl 277	. . . . . . . 8 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
25	22, 24	rexbid 2474	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝑧 = 𝐴) → (∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
26	25	adantll 476	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → (∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
27	19, 26	bitrid 192	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
28	11, 27	sbcied 2997	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
29	10, 28	bitr3id 194	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
30	29	expcom 116	. 2 ⊢ (Ⅎ𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
31	2, 9, 30	pm5.21ndd 705	1 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  [wsb 1760   ∈ wcel 2146  Ⅎwnfc 2304  ∃wrex 2454  Vcvv 2735  [wsbc 2960

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961

This theorem is referenced by:  sbcrex  3040