Theorem ralxpf 5268

Description: Version of ralxp 5263 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 3182	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑)
2		cbvralsv 3182	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑣 ∈ 𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
3	2	ralbii 2980	. . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑧 ∈ 𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
4		nfv 1843	. . . 4 ⊢ Ⅎ𝑢∀𝑧 ∈ 𝐵 𝜓
5		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦𝐵
6		nfs1v 2437	. . . . 5 ⊢ Ⅎ𝑦[𝑢 / 𝑦]𝜓
7	5, 6	nfral 2945	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝐵 [𝑢 / 𝑦]𝜓
8		sbequ12 2111	. . . . 5 ⊢ (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
9	8	ralbidv 2986	. . . 4 ⊢ (𝑦 = 𝑢 → (∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑧 ∈ 𝐵 [𝑢 / 𝑦]𝜓))
10	4, 7, 9	cbvral 3167	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑢 ∈ 𝐴 ∀𝑧 ∈ 𝐵 [𝑢 / 𝑦]𝜓)
11		vex 3203	. . . . . 6 ⊢ 𝑢 ∈ V
12		vex 3203	. . . . . 6 ⊢ 𝑣 ∈ V
13	11, 12	eqvinop 4955	. . . . 5 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩))
14		ralxpf.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
15	14	nfsb 2440	. . . . . . 7 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
16	6	nfsb 2440	. . . . . . 7 ⊢ Ⅎ𝑦[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
17	15, 16	nfbi 1833	. . . . . 6 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
18		ralxpf.2	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
19	18	nfsb 2440	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
20		nfs1v 2437	. . . . . . . 8 ⊢ Ⅎ𝑧[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
21	19, 20	nfbi 1833	. . . . . . 7 ⊢ Ⅎ𝑧([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
22		ralxpf.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
23		ralxpf.4	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
24	22, 23	sbhypf 3253	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ([𝑤 / 𝑥]𝜑 ↔ 𝜓))
25		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26		vex 3203	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	25, 26	opth 4945	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑦 = 𝑢 ∧ 𝑧 = 𝑣))
28		sbequ12 2111	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ([𝑢 / 𝑦]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
29	8, 28	sylan9bb 736	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
30	27, 29	sylbi 207	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩ → (𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
31	24, 30	sylan9bb 736	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
32	21, 31	exlimi 2086	. . . . . 6 ⊢ (∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
33	17, 32	exlimi 2086	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
34	13, 33	sylbi 207	. . . 4 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
35	34	ralxp 5263	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
36	3, 10, 35	3bitr4ri 293	. 2 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
37	1, 36	bitri 264	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704  Ⅎwnf 1708  [wsb 1880  ∀wral 2912  ⟨cop 4183   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  rexxpf  5269