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Theorem rexiunxp 4751
Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 4753, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
ralxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
rexiunxp (∃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 rexiunxp
StepHypRef Expression
1 eliunxp 4748 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21anbi1i 455 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
3 19.41vv 1896 . . . . 5 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
42, 3bitr4i 186 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
54exbii 1598 . . 3 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
6 exrot3 1683 . . . 4 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
7 anass 399 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
87exbii 1598 . . . . . 6 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
9 vex 2733 . . . . . . . 8 𝑦 ∈ V
10 vex 2733 . . . . . . . 8 𝑧 ∈ V
119, 10opex 4212 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
12 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1312anbi2d 461 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓)))
1411, 13ceqsexv 2769 . . . . . 6 (∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
158, 14bitri 183 . . . . 5 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
16152exbii 1599 . . . 4 (∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
176, 16bitri 183 . . 3 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
185, 17bitri 183 . 2 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
19 df-rex 2454 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑))
20 r2ex 2490 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
2118, 19, 203bitr4i 211 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  wrex 2449  {csn 3581  cop 3584   ciun 3871   × cxp 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-iun 3873  df-opab 4049  df-xp 4615  df-rel 4616
This theorem is referenced by:  rexxp  4753
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