New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ralprg GIF version

Theorem ralprg 3775
 Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1 (x = A → (φψ))
ralprg.2 (x = B → (φχ))
Assertion
Ref Expression
ralprg ((A V B W) → (x {A, B}φ ↔ (ψ χ)))
Distinct variable groups:   x,A   x,B   ψ,x   χ,x
Allowed substitution hints:   φ(x)   V(x)   W(x)

Proof of Theorem ralprg
StepHypRef Expression
1 df-pr 3742 . . . 4 {A, B} = ({A} ∪ {B})
21raleqi 2811 . . 3 (x {A, B}φx ({A} ∪ {B})φ)
3 ralunb 3444 . . 3 (x ({A} ∪ {B})φ ↔ (x {A}φ x {B}φ))
42, 3bitri 240 . 2 (x {A, B}φ ↔ (x {A}φ x {B}φ))
5 ralprg.1 . . . 4 (x = A → (φψ))
65ralsng 3765 . . 3 (A V → (x {A}φψ))
7 ralprg.2 . . . 4 (x = B → (φχ))
87ralsng 3765 . . 3 (B W → (x {B}φχ))
96, 8bi2anan9 843 . 2 ((A V B W) → ((x {A}φ x {B}φ) ↔ (ψ χ)))
104, 9syl5bb 248 1 ((A V B W) → (x {A, B}φ ↔ (ψ χ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  raltpg  3777  ralpr  3779  iinxprg  4043
 Copyright terms: Public domain W3C validator