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Theorem raliunxp 4505
Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4507, 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 4503 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21imbi1i 231 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
3 19.23vv 1780 . . . . 5 (∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
42, 3bitr4i 180 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
54albii 1375 . . 3 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
6 alrot3 1390 . . . 4 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑))
7 impexp 254 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
87albii 1375 . . . . . 6 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)))
9 vex 2577 . . . . . . . 8 𝑦 ∈ V
10 vex 2577 . . . . . . . 8 𝑧 ∈ V
119, 10opex 3994 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
12 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1312imbi2d 223 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓)))
1411, 13ceqsalv 2601 . . . . . 6 (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐴𝑧𝐵) → 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
158, 14bitri 177 . . . . 5 (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ((𝑦𝐴𝑧𝐵) → 𝜓))
16152albii 1376 . . . 4 (∀𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
176, 16bitri 177 . . 3 (∀𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
185, 17bitri 177 . 2 (∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
19 df-ral 2328 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) → 𝜑))
20 r2al 2360 . 2 (∀𝑦𝐴𝑧𝐵 𝜓 ↔ ∀𝑦𝑧((𝑦𝐴𝑧𝐵) → 𝜓))
2118, 19, 203bitr4i 205 1 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  wral 2323  {csn 3403  cop 3406   ciun 3685   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-iun 3687  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by:  ralxp  4507  fmpt2x  5854
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