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Theorem 2ralunsn 3597
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 3596 . . 3 (𝐵𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦𝐴 𝜑𝜓)))
32ralbidv 2343 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓)))
4 2ralunsn.1 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜒))
54ralbidv 2343 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝐴 𝜑 ↔ ∀𝑦𝐴 𝜒))
6 2ralunsn.3 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜃))
75, 6anbi12d 450 . . . 4 (𝑥 = 𝐵 → ((∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑦𝐴 𝜒𝜃)))
87ralunsn 3596 . . 3 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
9 r19.26 2458 . . . 4 (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
109anbi1i 439 . . 3 ((∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)))
118, 10syl6bb 189 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
123, 11bitrd 181 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 101  wb 102   = wceq 1259  wcel 1409  wral 2323  cun 2943  {csn 3403
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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
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-v 2576  df-sbc 2788  df-un 2950  df-sn 3409
This theorem is referenced by: (None)
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