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Theorem r19.9rzv 3644
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.9rzv (A → (φx A φ))
Distinct variable groups:   x,A   φ,x

Proof of Theorem r19.9rzv
StepHypRef Expression
1 r19.3rzv 3643 . . . 4 (A → (¬ φx A ¬ φ))
21bicomd 192 . . 3 (A → (x A ¬ φ ↔ ¬ φ))
32con2bid 319 . 2 (A → (φ ↔ ¬ x A ¬ φ))
4 dfrex2 2627 . 2 (x A φ ↔ ¬ x A ¬ φ)
53, 4syl6bbr 254 1 (A → (φx A φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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:  r19.45zv  3647  r19.36zv  3650  iunconst  3977  fconstfv  5456
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