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Theorem raliunxp 5800
Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 5802, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
raliunxp (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝐵(𝑦)

Proof of Theorem raliunxp
StepHypRef Expression
1 eliunxp 5798 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21imbi1i 350 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
3 19.23vv 1947 . . . . 5 (∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
42, 3bitr4i 278 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
54albii 1822 . . 3 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
6 alrot3 2158 . . . 4 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
7 impexp 452 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
87albii 1822 . . . . . 6 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
9 opex 5426 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
10 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1110imbi2d 341 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓)))
129, 11ceqsalv 3484 . . . . . 6 (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
138, 12bitri 275 . . . . 5 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
14132albii 1823 . . . 4 (∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
156, 14bitri 275 . . 3 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
165, 15bitri 275 . 2 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
17 df-ral 3066 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑))
18 r2al 3192 . 2 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
1916, 17, 183bitr4i 303 1 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wral 3065  {csn 4591  cop 4597   ciun 4959   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4961  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  rexiunxp  5801  ralxp  5802  fmpox  8004  ovmptss  8030  filnetlem4  34882
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