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Theorem ralxpf 5406
Description: Version of ralxp 5401 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 3331 . 2 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑤 ∈ (𝐴 × 𝐵)[𝑤 / 𝑥]𝜑)
2 cbvralsv 3331 . . . 4 (∀𝑧𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑣𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
32ralbii 3129 . . 3 (∀𝑢𝐴𝑧𝐵 [𝑢 / 𝑦]𝜓 ↔ ∀𝑢𝐴𝑣𝐵 [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
4 nfv 1995 . . . 4 𝑢𝑧𝐵 𝜓
5 nfcv 2913 . . . . 5 𝑦𝐵
6 nfs1v 2274 . . . . 5 𝑦[𝑢 / 𝑦]𝜓
75, 6nfral 3094 . . . 4 𝑦𝑧𝐵 [𝑢 / 𝑦]𝜓
8 sbequ12 2267 . . . . 5 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
98ralbidv 3135 . . . 4 (𝑦 = 𝑢 → (∀𝑧𝐵 𝜓 ↔ ∀𝑧𝐵 [𝑢 / 𝑦]𝜓))
104, 7, 9cbvral 3316 . . 3 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑢𝐴𝑧𝐵 [𝑢 / 𝑦]𝜓)
11 vex 3354 . . . . . 6 𝑢 ∈ V
12 vex 3354 . . . . . 6 𝑣 ∈ V
1311, 12eqvinop 5083 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩))
14 ralxpf.1 . . . . . . . 8 𝑦𝜑
1514nfsb 2590 . . . . . . 7 𝑦[𝑤 / 𝑥]𝜑
166nfsb 2590 . . . . . . 7 𝑦[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
1715, 16nfbi 1985 . . . . . 6 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓)
18 ralxpf.2 . . . . . . . . 9 𝑧𝜑
1918nfsb 2590 . . . . . . . 8 𝑧[𝑤 / 𝑥]𝜑
20 nfs1v 2274 . . . . . . . 8 𝑧[𝑣 / 𝑧][𝑢 / 𝑦]𝜓
2119, 20nfbi 1985 . . . . . . 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 3405 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑧⟩ → ([𝑤 / 𝑥]𝜑𝜓))
25 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
26 vex 3354 . . . . . . . . . 10 𝑧 ∈ V
2725, 26opth 5073 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑦 = 𝑢𝑧 = 𝑣))
28 sbequ12 2267 . . . . . . . . . 10 (𝑧 = 𝑣 → ([𝑢 / 𝑦]𝜓 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
298, 28sylan9bb 499 . . . . . . . . 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 499 . . . . . . 7 ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3221, 31exlimi 2242 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ = ⟨𝑢, 𝑣⟩) → ([𝑤 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑢 / 𝑦]𝜓))
3317, 32exlimi 2242 . . . . 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 5401 . . 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 382   = wceq 1631  wex 1852  wnf 1856  [wsb 2049  wral 3061  cop 4323   × cxp 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-iun 4657  df-opab 4848  df-xp 5256  df-rel 5257
This theorem is referenced by:  rexxpf  5407
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