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Theorem rexxpf 4541
Description: Version of rexxp 4538 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (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
rexxpf (∃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 rexxpf
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvrexsv 2595 . 2 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2 cbvrexsv 2595 . . . 4 (∃𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
32rexbii 2379 . . 3 (∃𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4 nfv 1462 . . . 4 𝑤𝑧𝐵 𝜓
5 nfcv 2223 . . . . 5 𝑦𝐵
6 nfs1v 1858 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
75, 6nfrexxy 2409 . . . 4 𝑦𝑧𝐵 [𝑤 / 𝑦]𝜓
8 sbequ12 1696 . . . . 5 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
98rexbidv 2375 . . . 4 (𝑦 = 𝑤 → (∃𝑧𝐵 𝜓 ↔ ∃𝑧𝐵 [𝑤 / 𝑦]𝜓))
104, 7, 9cbvrex 2580 . . 3 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ∃𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓)
11 vex 2615 . . . . . 6 𝑤 ∈ V
12 vex 2615 . . . . . 6 𝑢 ∈ V
1311, 12eqvinop 4034 . . . . 5 (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 1865 . . . . . . 7 𝑦[𝑣 / 𝑥]𝜑
166nfsb 1865 . . . . . . 7 𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
1715, 16nfbi 1522 . . . . . 6 𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 1865 . . . . . . . 8 𝑧[𝑣 / 𝑥]𝜑
20 nfs1v 1858 . . . . . . . 8 𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
2119, 20nfbi 1522 . . . . . . 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 2659 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑𝜓))
25 vex 2615 . . . . . . . . . 10 𝑦 ∈ V
26 vex 2615 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 4028 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤𝑧 = 𝑢))
28 sbequ12 1696 . . . . . . . . . 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 1526 . . . . . 6 (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3317, 32exlimi 1526 . . . . 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))
3534rexxp 4538 . . 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 1285  wnf 1390  wex 1422  [wsb 1687  wrex 2354  cop 3425   × cxp 4399
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-iun 3706  df-opab 3866  df-xp 4407  df-rel 4408
This theorem is referenced by:  iunxpf  4542
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