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Theorem ralxpf 5301
Description: Version of ralxp 5296 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
ralxpf.1 𝑦𝜑
ralxpf.2 𝑧𝜑
ralxpf.3 𝑥𝜓
ralxpf.4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
ralxpf (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 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.
StepHypRef Expression
1 cbvralsv 3212 . 2 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑)
2 cbvralsv 3212 . . . 4 (∀𝑧𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑣𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
32ralbii 3009 . . 3 (∀𝑢𝐴𝑧𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑢𝐴𝑣𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
4 nfv 1883 . . . 4 𝑢𝑧𝐵 𝜓
5 nfcv 2793 . . . . 5 𝑦𝐵
6 nfs1v 2465 . . . . 5 𝑦[𝑢 / 𝑦]𝜓
75, 6nfral 2974 . . . 4 𝑦𝑧𝐵 [𝑢 / 𝑦]𝜓
8 sbequ12 2149 . . . . 5 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
98ralbidv 3015 . . . 4 (𝑦 = 𝑢 → (∀𝑧𝐵 𝜓 ↔ ∀𝑧𝐵 [𝑢 / 𝑦]𝜓))
104, 7, 9cbvral 3197 . . 3 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑢𝐴𝑧𝐵 [𝑢 / 𝑦]𝜓)
11 vex 3234 . . . . . 6 𝑢 ∈ V
12 vex 3234 . . . . . 6 𝑣 ∈ V
1311, 12eqvinop 4984 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 2468 . . . . . . 7 𝑦[𝑤 / 𝑥]𝜑
166nfsb 2468 . . . . . . 7 𝑦[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
1715, 16nfbi 1873 . . . . . 6 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 2468 . . . . . . . 8 𝑧[𝑤 / 𝑥]𝜑
20 nfs1v 2465 . . . . . . . 8 𝑧[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
2119, 20nfbi 1873 . . . . . . 7 𝑧([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
22 ralxpf.3 . . . . . . . . 9 𝑥𝜓
23 ralxpf.4 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
2422, 23sbhypf 3284 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑧⟩ → ([𝑤 / 𝑥]𝜑𝜓))
25 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
26 vex 3234 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 4974 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑦 = 𝑢𝑧 = 𝑣))
28 sbequ12 2149 . . . . . . . . . 10 (𝑧 = 𝑣 → ([𝑢 / 𝑦]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
298, 28sylan9bb 736 . . . . . . . . 9 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3027, 29sylbi 207 . . . . . . . 8 (⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩ → (𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3124, 30sylan9bb 736 . . . . . . 7 ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3221, 31exlimi 2124 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3317, 32exlimi 2124 . . . . 5 (∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3413, 33sylbi 207 . . . 4 (𝑤 = ⟨𝑢, 𝑣⟩ → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3534ralxp 5296 . . 3 (∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑 ↔ ∀𝑢𝐴𝑣𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
363, 10, 353bitr4ri 293 . 2 (∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
371, 36bitri 264 1 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wnf 1748  [wsb 1937  wral 2941  cop 4216   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  rexxpf  5302
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