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Theorem r19.23t 3169
Description: Closed theorem form of r19.23 3170. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
r19.23t (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Proof of Theorem r19.23t
StepHypRef Expression
1 19.23t 2235 . 2 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓)))
2 df-ral 3066 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 impexp 437 . . . 4 (((𝑥𝐴𝜑) → 𝜓) ↔ (𝑥𝐴 → (𝜑𝜓)))
43albii 1895 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
52, 4bitr4i 267 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝜓))
6 df-rex 3067 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76imbi1i 338 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓))
81, 5, 73bitr4g 303 1 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852  wnf 1856  wcel 2145  wral 3061  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-ral 3066  df-rex 3067
This theorem is referenced by:  r19.23  3170  rexlimd2  3173  riotasv3d  34768
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