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Theorem ralxpf 4732
Description: Version of ralxp 4729 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 2694 . 2 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2 cbvralsv 2694 . . . 4 (∀𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
32ralbii 2463 . . 3 (∀𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∀𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4 nfv 1508 . . . 4 𝑤𝑧𝐵 𝜓
5 nfcv 2299 . . . . 5 𝑦𝐵
6 nfs1v 1919 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
75, 6nfralxy 2495 . . . 4 𝑦𝑧𝐵 [𝑤 / 𝑦]𝜓
8 sbequ12 1751 . . . . 5 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
98ralbidv 2457 . . . 4 (𝑦 = 𝑤 → (∀𝑧𝐵 𝜓 ↔ ∀𝑧𝐵 [𝑤 / 𝑦]𝜓))
104, 7, 9cbvral 2676 . . 3 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓)
11 vex 2715 . . . . . 6 𝑤 ∈ V
12 vex 2715 . . . . . 6 𝑢 ∈ V
1311, 12eqvinop 4203 . . . . 5 (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 1926 . . . . . . 7 𝑦[𝑣 / 𝑥]𝜑
166nfsb 1926 . . . . . . 7 𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
1715, 16nfbi 1569 . . . . . 6 𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 1926 . . . . . . . 8 𝑧[𝑣 / 𝑥]𝜑
20 nfs1v 1919 . . . . . . . 8 𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
2119, 20nfbi 1569 . . . . . . 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 2761 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑𝜓))
25 vex 2715 . . . . . . . . . 10 𝑦 ∈ V
26 vex 2715 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 4197 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤𝑧 = 𝑢))
28 sbequ12 1751 . . . . . . . . . 10 (𝑧 = 𝑢 → ([𝑤 / 𝑦]𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
298, 28sylan9bb 458 . . . . . . . . 9 ((𝑦 = 𝑤𝑧 = 𝑢) → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3027, 29sylbi 120 . . . . . . . 8 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ → (𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3124, 30sylan9bb 458 . . . . . . 7 ((𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3221, 31exlimi 1574 . . . . . 6 (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3317, 32exlimi 1574 . . . . 5 (∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3413, 33sylbi 120 . . . 4 (𝑣 = ⟨𝑤, 𝑢⟩ → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3534ralxp 4729 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
363, 10, 353bitr4ri 212 . 2 (∀𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
371, 36bitri 183 1 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wnf 1440  wex 1472  [wsb 1742  wral 2435  cop 3563   × cxp 4584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-iun 3851  df-opab 4026  df-xp 4592  df-rel 4593
This theorem is referenced by: (None)
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