Theorem rexxpf 4614

Description: Version of rexxp 4611 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
ralxpf.1	⊢ Ⅎ𝑦𝜑
ralxpf.2	⊢ Ⅎ𝑧𝜑
ralxpf.3	⊢ Ⅎ𝑥𝜓
ralxpf.4	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexxpf	⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem rexxpf

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvrexsv 2616	. 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2		cbvrexsv 2616	. . . 4 ⊢ (∃𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
3	2	rexbii 2396	. . 3 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑤 ∈ 𝐴 ∃𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4		nfv 1473	. . . 4 ⊢ Ⅎ𝑤∃𝑧 ∈ 𝐵 𝜓
5		nfcv 2235	. . . . 5 ⊢ Ⅎ𝑦𝐵
6		nfs1v 1870	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
7	5, 6	nfrexxy 2426	. . . 4 ⊢ Ⅎ𝑦∃𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓
8		sbequ12 1708	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
9	8	rexbidv 2392	. . . 4 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐵 𝜓 ↔ ∃𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓))
10	4, 7, 9	cbvrex 2601	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓)
11		vex 2636	. . . . . 6 ⊢ 𝑤 ∈ V
12		vex 2636	. . . . . 6 ⊢ 𝑢 ∈ V
13	11, 12	eqvinop 4094	. . . . 5 ⊢ (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14		ralxpf.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
15	14	nfsb 1877	. . . . . . 7 ⊢ Ⅎ𝑦[𝑣 / 𝑥]𝜑
16	6	nfsb 1877	. . . . . . 7 ⊢ Ⅎ𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
17	15, 16	nfbi 1533	. . . . . 6 ⊢ Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18		ralxpf.2	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
19	18	nfsb 1877	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑣 / 𝑥]𝜑
20		nfs1v 1870	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
21	19, 20	nfbi 1533	. . . . . . 7 ⊢ Ⅎ𝑧([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
22		ralxpf.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
23		ralxpf.4	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
24	22, 23	sbhypf 2682	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑 ↔ 𝜓))
25		vex 2636	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26		vex 2636	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	25, 26	opth 4088	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤 ∧ 𝑧 = 𝑢))
28		sbequ12 1708	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ([𝑤 / 𝑦]𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
29	8, 28	sylan9bb 451	. . . . . . . . 9 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑢) → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
30	27, 29	sylbi 120	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
31	24, 30	sylan9bb 451	. . . . . . 7 ⊢ ((𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
32	21, 31	exlimi 1537	. . . . . 6 ⊢ (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
33	17, 32	exlimi 1537	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
34	13, 33	sylbi 120	. . . 4 ⊢ (𝑣 = ⟨𝑤, 𝑢⟩ → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
35	34	rexxp 4611	. . 3 ⊢ (∃𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∃𝑤 ∈ 𝐴 ∃𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
36	3, 10, 35	3bitr4ri 212	. 2 ⊢ (∃𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
37	1, 36	bitri 183	1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  Ⅎwnf 1401  ∃wex 1433  [wsb 1699  ∃wrex 2371  ⟨cop 3469   × cxp 4465

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-iun 3754  df-opab 3922  df-xp 4473  df-rel 4474

This theorem is referenced by:  iunxpf  4615