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Theorem raliunxp 5231
Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 5233, 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 5229 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21imbi1i 339 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
3 19.23vv 1900 . . . . 5 (∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
42, 3bitr4i 267 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
54albii 1744 . . 3 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
6 alrot3 2035 . . . 4 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
7 impexp 462 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
87albii 1744 . . . . . 6 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
9 opex 4903 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
10 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1110imbi2d 330 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓)))
129, 11ceqsalv 3223 . . . . . 6 (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
138, 12bitri 264 . . . . 5 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
14132albii 1745 . . . 4 (∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
156, 14bitri 264 . . 3 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
165, 15bitri 264 . 2 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
17 df-ral 2913 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑))
18 r2al 2935 . 2 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
1916, 17, 183bitr4i 292 1 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2908  {csn 4155  cop 4161   ciun 4492   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-iun 4494  df-opab 4684  df-xp 5090  df-rel 5091
This theorem is referenced by:  rexiunxp  5232  ralxp  5233  fmpt2x  7196  ovmptss  7218  filnetlem4  32071
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