New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  reximdva0 GIF version

Theorem reximdva0 3561
 Description: Restricted existence deduced from non-empty class. (Contributed by NM, 1-Feb-2012.)
Hypothesis
Ref Expression
reximdva0.1 ((φ x A) → ψ)
Assertion
Ref Expression
reximdva0 ((φ A) → x A ψ)
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem reximdva0
StepHypRef Expression
1 n0 3559 . . 3 (Ax x A)
2 reximdva0.1 . . . . . . 7 ((φ x A) → ψ)
32ex 423 . . . . . 6 (φ → (x Aψ))
43ancld 536 . . . . 5 (φ → (x A → (x A ψ)))
54eximdv 1622 . . . 4 (φ → (x x Ax(x A ψ)))
65imp 418 . . 3 ((φ x x A) → x(x A ψ))
71, 6sylan2b 461 . 2 ((φ A) → x(x A ψ))
8 df-rex 2620 . 2 (x A ψx(x A ψ))
97, 8sylibr 203 1 ((φ A) → x A ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550 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-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-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator