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Theorem ralprgf 4606
 Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 8-Apr-2023.)
Hypotheses
Ref Expression
ralprgf.1 𝑥𝜓
ralprgf.2 𝑥𝜒
ralprgf.a (𝑥 = 𝐴 → (𝜑𝜓))
ralprgf.b (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
ralprgf ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ralprgf
StepHypRef Expression
1 df-pr 4546 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21raleqi 3396 . . 3 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 ralunb 4146 . . 3 (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
42, 3bitri 277 . 2 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5 ralprgf.1 . . . 4 𝑥𝜓
6 ralprgf.a . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6ralsngf 4587 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
8 ralprgf.2 . . . 4 𝑥𝜒
9 ralprgf.b . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
108, 9ralsngf 4587 . . 3 (𝐵𝑊 → (∀𝑥 ∈ {𝐵}𝜑𝜒))
117, 10bi2anan9 637 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
124, 11syl5bb 285 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3125   ∪ cun 3911  {csn 4543  {cpr 4545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ral 3130  df-v 3475  df-sbc 3753  df-un 3918  df-sn 4544  df-pr 4546 This theorem is referenced by:  ralprg  4608
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