New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  2ralbida GIF version

Theorem 2ralbida 2653
 Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
Hypotheses
Ref Expression
2ralbida.1 xφ
2ralbida.2 yφ
2ralbida.3 ((φ (x A y B)) → (ψχ))
Assertion
Ref Expression
2ralbida (φ → (x A y B ψx A y B χ))
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)   A(x)   B(x,y)

Proof of Theorem 2ralbida
StepHypRef Expression
1 2ralbida.1 . 2 xφ
2 2ralbida.2 . . . 4 yφ
3 nfv 1619 . . . 4 y x A
42, 3nfan 1824 . . 3 y(φ x A)
5 2ralbida.3 . . . 4 ((φ (x A y B)) → (ψχ))
65anassrs 629 . . 3 (((φ x A) y B) → (ψχ))
74, 6ralbida 2628 . 2 ((φ x A) → (y B ψy B χ))
81, 7ralbida 2628 1 (φ → (x A y B ψx A y B χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2614 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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  2ralbidva  2654
 Copyright terms: Public domain W3C validator