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Theorem raliunxp 4786
Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4788, 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 4784 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21imbi1i 238 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
3 19.23vv 1895 . . . . 5 (∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
42, 3bitr4i 187 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
54albii 1481 . . 3 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
6 alrot3 1496 . . . 4 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
7 impexp 263 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
87albii 1481 . . . . . 6 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
9 vex 2755 . . . . . . . 8 𝑦 ∈ V
10 vex 2755 . . . . . . . 8 𝑧 ∈ V
119, 10opex 4247 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
12 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1312imbi2d 230 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓)))
1411, 13ceqsalv 2782 . . . . . 6 (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
158, 14bitri 184 . . . . 5 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
16152albii 1482 . . . 4 (∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
176, 16bitri 184 . . 3 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
185, 17bitri 184 . 2 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
19 df-ral 2473 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑))
20 r2al 2509 . 2 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
2118, 19, 203bitr4i 212 1 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  wral 2468  {csn 3607  cop 3610   ciun 3901   × cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iun 3903  df-opab 4080  df-xp 4650  df-rel 4651
This theorem is referenced by:  ralxp  4788  fmpox  6225
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