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

Theorem ralpr 3779
 Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1 A V
ralpr.2 B V
ralpr.3 (x = A → (φψ))
ralpr.4 (x = B → (φχ))
Assertion
Ref Expression
ralpr (x {A, B}φ ↔ (ψ χ))
Distinct variable groups:   x,A   x,B   ψ,x   χ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2 A V
2 ralpr.2 . 2 B V
3 ralpr.3 . . 3 (x = A → (φψ))
4 ralpr.4 . . 3 (x = B → (φχ))
53, 4ralprg 3775 . 2 ((A V B V) → (x {A, B}φ ↔ (ψ χ)))
61, 2, 5mp2an 653 1 (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  Vcvv 2859  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
 Copyright terms: Public domain W3C validator