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Theorem 2ralunsn 3796
Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
Hypotheses
Ref Expression
2ralunsn.1 (𝑥 = 𝐵 → (𝜑𝜒))
2ralunsn.2 (𝑦 = 𝐵 → (𝜑𝜓))
2ralunsn.3 (𝑥 = 𝐵 → (𝜓𝜃))
Assertion
Ref Expression
2ralunsn (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝜒,𝑥   𝜓,𝑦   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem 2ralunsn
StepHypRef Expression
1 2ralunsn.2 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
21ralunsn 3795 . . 3 (𝐵𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦𝐴 𝜑𝜓)))
32ralbidv 2477 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓)))
4 2ralunsn.1 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜒))
54ralbidv 2477 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝐴 𝜑 ↔ ∀𝑦𝐴 𝜒))
6 2ralunsn.3 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜃))
75, 6anbi12d 473 . . . 4 (𝑥 = 𝐵 → ((∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑦𝐴 𝜒𝜃)))
87ralunsn 3795 . . 3 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
9 r19.26 2603 . . . 4 (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
109anbi1i 458 . . 3 ((∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)))
118, 10bitrdi 196 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
123, 11bitrd 188 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cun 3127  {csn 3591
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3597
This theorem is referenced by: (None)
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