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Theorem raliunxp 5183
 Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 5185, 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 5181 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21imbi1i 338 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
3 19.23vv 1890 . . . . 5 (∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
42, 3bitr4i 266 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
54albii 1737 . . 3 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
6 alrot3 2025 . . . 4 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
7 impexp 461 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
87albii 1737 . . . . . 6 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
9 opex 4859 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
10 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1110imbi2d 329 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓)))
129, 11ceqsalv 3206 . . . . . 6 (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
138, 12bitri 263 . . . . 5 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
14132albii 1738 . . . 4 (∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
156, 14bitri 263 . . 3 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
165, 15bitri 263 . 2 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
17 df-ral 2901 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑))
18 r2al 2923 . 2 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
1916, 17, 183bitr4i 291 1 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by:  rexiunxp  5184  ralxp  5185  fmpt2x  7125  ovmptss  7145  filnetlem4  31546
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