New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  r19.37zv GIF version

Theorem r19.37zv 3646
 Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
Assertion
Ref Expression
r19.37zv (A → (x A (φψ) ↔ (φx A ψ)))
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem r19.37zv
StepHypRef Expression
1 r19.3rzv 3643 . . 3 (A → (φx A φ))
21imbi1d 308 . 2 (A → ((φx A ψ) ↔ (x A φx A ψ)))
3 r19.35 2758 . 2 (x A (φψ) ↔ (x A φx A ψ))
42, 3syl6rbbr 255 1 (A → (x A (φψ) ↔ (φx A ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∅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-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:  ncssfin  6151
 Copyright terms: Public domain W3C validator