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Theorem ralprgf 4691
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 4626 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21raleqi 3317 . . 3 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 ralunb 4186 . . 3 (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
42, 3bitri 275 . 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 4670 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
8 ralprgf.2 . . . 4 𝑥𝜒
9 ralprgf.b . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
108, 9ralsngf 4670 . . 3 (𝐵𝑊 → (∀𝑥 ∈ {𝐵}𝜑𝜒))
117, 10bi2anan9 636 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
124, 11bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wnf 1777  wcel 2098  wral 3055  cun 3941  {csn 4623  {cpr 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by:  ralprgOLD  4694
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