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Theorem ralxpf 4570
Description: Version of ralxp 4567 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 2601 . 2 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2 cbvralsv 2601 . . . 4 (∀𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
32ralbii 2384 . . 3 (∀𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4 nfv 1466 . . . 4 𝑤𝑧𝐵 𝜓
5 nfcv 2228 . . . . 5 𝑦𝐵
6 nfs1v 1863 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
75, 6nfralxy 2414 . . . 4 𝑦𝑧𝐵 [𝑤 / 𝑦]𝜓
8 sbequ12 1701 . . . . 5 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
98ralbidv 2380 . . . 4 (𝑦 = 𝑤 → (∀𝑧𝐵 𝜓 ↔ ∀𝑧𝐵 [𝑤 / 𝑦]𝜓))
104, 7, 9cbvral 2586 . . 3 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓)
11 vex 2622 . . . . . 6 𝑤 ∈ V
12 vex 2622 . . . . . 6 𝑢 ∈ V
1311, 12eqvinop 4061 . . . . 5 (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 1870 . . . . . . 7 𝑦[𝑣 / 𝑥]𝜑
166nfsb 1870 . . . . . . 7 𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
1715, 16nfbi 1526 . . . . . 6 𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 1870 . . . . . . . 8 𝑧[𝑣 / 𝑥]𝜑
20 nfs1v 1863 . . . . . . . 8 𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
2119, 20nfbi 1526 . . . . . . 7 𝑧([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
22 ralxpf.3 . . . . . . . . 9 𝑥𝜓
23 ralxpf.4 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
2422, 23sbhypf 2668 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑𝜓))
25 vex 2622 . . . . . . . . . 10 𝑦 ∈ V
26 vex 2622 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 4055 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤𝑧 = 𝑢))
28 sbequ12 1701 . . . . . . . . . 10 (𝑧 = 𝑢 → ([𝑤 / 𝑦]𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
298, 28sylan9bb 450 . . . . . . . . 9 ((𝑦 = 𝑤𝑧 = 𝑢) → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3027, 29sylbi 119 . . . . . . . 8 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3124, 30sylan9bb 450 . . . . . . 7 ((𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3221, 31exlimi 1530 . . . . . 6 (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3317, 32exlimi 1530 . . . . 5 (∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3413, 33sylbi 119 . . . 4 (𝑣 = ⟨𝑤, 𝑢⟩ → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3534ralxp 4567 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
363, 10, 353bitr4ri 211 . 2 (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
371, 36bitri 182 1 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wnf 1394  wex 1426  [wsb 1692  wral 2359  cop 3444   × cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-iun 3727  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by: (None)
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