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Theorem 2ralunsn 4575
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 4574 . . 3 (𝐵𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦𝐴 𝜑𝜓)))
32ralbidv 3124 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓)))
4 2ralunsn.1 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜒))
54ralbidv 3124 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝐴 𝜑 ↔ ∀𝑦𝐴 𝜒))
6 2ralunsn.3 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜃))
75, 6anbi12d 749 . . . 4 (𝑥 = 𝐵 → ((∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑦𝐴 𝜒𝜃)))
87ralunsn 4574 . . 3 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
9 r19.26 3202 . . . 4 (∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
109anbi1i 733 . . 3 ((∀𝑥𝐴 (∀𝑦𝐴 𝜑𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃)))
118, 10syl6bb 276 . 2 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦𝐴 𝜑𝜓) ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
123, 11bitrd 268 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cun 3713  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-sbc 3577  df-un 3720  df-sn 4322
This theorem is referenced by: (None)
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