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Theorem rexxpf 4614
 Description: Version of rexxp 4611 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 2616 . 2 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑣 ∈ (𝐴 × 𝐵)[𝑣 / 𝑥]𝜑)
2 cbvrexsv 2616 . . . 4 (∃𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
32rexbii 2396 . . 3 (∃𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓 ↔ ∃𝑤𝐴𝑢𝐵 [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
4 nfv 1473 . . . 4 𝑤𝑧𝐵 𝜓
5 nfcv 2235 . . . . 5 𝑦𝐵
6 nfs1v 1870 . . . . 5 𝑦[𝑤 / 𝑦]𝜓
75, 6nfrexxy 2426 . . . 4 𝑦𝑧𝐵 [𝑤 / 𝑦]𝜓
8 sbequ12 1708 . . . . 5 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
98rexbidv 2392 . . . 4 (𝑦 = 𝑤 → (∃𝑧𝐵 𝜓 ↔ ∃𝑧𝐵 [𝑤 / 𝑦]𝜓))
104, 7, 9cbvrex 2601 . . 3 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ∃𝑤𝐴𝑧𝐵 [𝑤 / 𝑦]𝜓)
11 vex 2636 . . . . . 6 𝑤 ∈ V
12 vex 2636 . . . . . 6 𝑢 ∈ V
1311, 12eqvinop 4094 . . . . 5 (𝑣 = ⟨𝑤, 𝑢⟩ ↔ ∃𝑦𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 1877 . . . . . . 7 𝑦[𝑣 / 𝑥]𝜑
166nfsb 1877 . . . . . . 7 𝑦[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
1715, 16nfbi 1533 . . . . . 6 𝑦([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 1877 . . . . . . . 8 𝑧[𝑣 / 𝑥]𝜑
20 nfs1v 1870 . . . . . . . 8 𝑧[𝑢 / 𝑧][𝑤 / 𝑦]𝜓
2119, 20nfbi 1533 . . . . . . 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 2682 . . . . . . . 8 (𝑣 = ⟨𝑦, 𝑧⟩ → ([𝑣 / 𝑥]𝜑𝜓))
25 vex 2636 . . . . . . . . . 10 𝑦 ∈ V
26 vex 2636 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 4088 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩ ↔ (𝑦 = 𝑤𝑧 = 𝑢))
28 sbequ12 1708 . . . . . . . . . 10 (𝑧 = 𝑢 → ([𝑤 / 𝑦]𝜓 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
298, 28sylan9bb 451 . . . . . . . . 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 451 . . . . . . 7 ((𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3221, 31exlimi 1537 . . . . . 6 (∃𝑧(𝑣 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑢⟩) → ([𝑣 / 𝑥]𝜑 ↔ [𝑢 / 𝑧][𝑤 / 𝑦]𝜓))
3317, 32exlimi 1537 . . . . 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))
3534rexxp 4611 . . 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 1296  Ⅎwnf 1401  ∃wex 1433  [wsb 1699  ∃wrex 2371  ⟨cop 3469   × cxp 4465 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-iun 3754  df-opab 3922  df-xp 4473  df-rel 4474 This theorem is referenced by:  iunxpf  4615
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