Theorem ralxpf 4921

Description: Version of ralxp 4918 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (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
ralxpf	⊢ (∀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 ralxpf

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

Step	Hyp	Ref	Expression
1		cbvralsv 2802	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2		cbvralsv 2802	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
3	2	ralbii 2556	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑤 ∈ 𝐴 ∀𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4		nfv 1581	. . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐵 𝜓
5		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑦𝐵
6		nfs1v 1999	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
7	5, 6	nfralxy 2588	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓
8		sbequ12 1824	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
9	8	ralbidv 2550	. . . 4 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓))
10	4, 7, 9	cbvral 2782	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐵 [𝑤 / 𝑦]𝜓)
11		vex 2824	. . . . . 6 ⊢ 𝑤 ∈ V
12		vex 2824	. . . . . 6 ⊢ 𝑢 ∈ V
13	11, 12	eqvinop 4378	. . . . 5 ⊢ (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14		ralxpf.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
15	14	nfsb 2006	. . . . . . 7 ⊢ Ⅎ𝑦[𝑣 / 𝑥]𝜑
16	6	nfsb 2006	. . . . . . 7 ⊢ Ⅎ𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
17	15, 16	nfbi 1642	. . . . . 6 ⊢ Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18		ralxpf.2	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
19	18	nfsb 2006	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑣 / 𝑥]𝜑
20		nfs1v 1999	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
21	19, 20	nfbi 1642	. . . . . . 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 2872	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑 ↔ 𝜓))
25		vex 2824	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26		vex 2824	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	25, 26	opth 4372	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤 ∧ 𝑧 = 𝑢))
28		sbequ12 1824	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ([𝑤 / 𝑦]𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
29	8, 28	sylan9bb 466	. . . . . . . . 9 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑢) → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
30	27, 29	sylbi 121	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
31	24, 30	sylan9bb 466	. . . . . . 7 ⊢ ((𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
32	21, 31	exlimi 1647	. . . . . 6 ⊢ (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
33	17, 32	exlimi 1647	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
34	13, 33	sylbi 121	. . . 4 ⊢ (𝑣 = ⟨𝑤, 𝑢⟩ → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
35	34	ralxp 4918	. . 3 ⊢ (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑢 ∈ 𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
36	3, 10, 35	3bitr4ri 213	. 2 ⊢ (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
37	1, 36	bitri 184	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  Ⅎwnf 1513  ∃wex 1545  [wsb 1815  ∀wral 2528  ⟨cop 3708   × cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-iun 4009  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by: (None)