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

Theorem ralimdaa 2691
 Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
Hypotheses
Ref Expression
ralimdaa.1 xφ
ralimdaa.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
ralimdaa (φ → (x A ψx A χ))

Proof of Theorem ralimdaa
StepHypRef Expression
1 ralimdaa.1 . . 3 xφ
2 ralimdaa.2 . . . . 5 ((φ x A) → (ψχ))
32ex 423 . . . 4 (φ → (x A → (ψχ)))
43a2d 23 . . 3 (φ → ((x Aψ) → (x Aχ)))
51, 4alimd 1764 . 2 (φ → (x(x Aψ) → x(x Aχ)))
6 df-ral 2619 . 2 (x A ψx(x Aψ))
7 df-ral 2619 . 2 (x A χx(x Aχ))
85, 6, 73imtr4g 261 1 (φ → (x A ψx A χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  ralimdva  2692
 Copyright terms: Public domain W3C validator