New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  r19.27z GIF version

Theorem r19.27z 3648
 Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
Hypothesis
Ref Expression
r19.27z.1 xψ
Assertion
Ref Expression
r19.27z (A → (x A (φ ψ) ↔ (x A φ ψ)))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem r19.27z
StepHypRef Expression
1 r19.27z.1 . . . 4 xψ
21r19.3rz 3641 . . 3 (A → (ψx A ψ))
32anbi2d 684 . 2 (A → ((x A φ ψ) ↔ (x A φ x A ψ)))
4 r19.26 2746 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
53, 4syl6rbbr 255 1 (A → (x A (φ ψ) ↔ (x A φ ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ≠ wne 2516  ∀wral 2614  ∅c0 3550 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-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-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  raaan  3657
 Copyright terms: Public domain W3C validator