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Theorem r19.23t 3230
Description: Closed theorem form of r19.23 3231. (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 2252 . 2 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓)))
2 df-ral 3122 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 impexp 443 . . . 4 (((𝑥𝐴𝜑) → 𝜓) ↔ (𝑥𝐴 → (𝜑𝜓)))
43albii 1918 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
52, 4bitr4i 270 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝜓))
6 df-rex 3123 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76imbi1i 341 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓))
81, 5, 73bitr4g 306 1 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1654  wex 1878  wnf 1882  wcel 2164  wral 3117  wrex 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-nf 1883  df-ral 3122  df-rex 3123
This theorem is referenced by:  r19.23  3231  rexlimd2  3234  riotasv3d  35028
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