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Theorem r19.2z 3639
 Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1659). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
Assertion
Ref Expression
r19.2z ((A x A φ) → x A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem r19.2z
StepHypRef Expression
1 df-ral 2619 . . . 4 (x A φx(x Aφ))
2 exintr 1614 . . . 4 (x(x Aφ) → (x x Ax(x A φ)))
31, 2sylbi 187 . . 3 (x A φ → (x x Ax(x A φ)))
4 n0 3559 . . 3 (Ax x A)
5 df-rex 2620 . . 3 (x A φx(x A φ))
63, 4, 53imtr4g 261 . 2 (x A φ → (Ax A φ))
76impcom 419 1 ((A x A φ) → x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃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-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:  r19.2zb  3640  intssuni  3948  riinn0  4040
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