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Theorem r19.3rz 3641
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
Hypothesis
Ref Expression
r19.3rz.1 xφ
Assertion
Ref Expression
r19.3rz (A → (φx A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem r19.3rz
StepHypRef Expression
1 n0 3559 . . 3 (Ax x A)
2 biimt 325 . . 3 (x x A → (φ ↔ (x x Aφ)))
31, 2sylbi 187 . 2 (A → (φ ↔ (x x Aφ)))
4 df-ral 2619 . . 3 (x A φx(x Aφ))
5 r19.3rz.1 . . . 4 xφ
6519.23 1801 . . 3 (x(x Aφ) ↔ (x x Aφ))
74, 6bitri 240 . 2 (x A φ ↔ (x x Aφ))
83, 7syl6bbr 254 1 (A → (φx A φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710   ≠ 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:  r19.28z  3642  r19.27z  3648
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