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Theorem r19.23t 3299
 Description: Closed theorem form of r19.23 3300. (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 2211 . 2 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓)))
2 df-ral 3135 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 impexp 454 . . . 4 (((𝑥𝐴𝜑) → 𝜓) ↔ (𝑥𝐴 → (𝜑𝜓)))
43albii 1821 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
52, 4bitr4i 281 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝜓))
6 df-rex 3136 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76imbi1i 353 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓))
81, 5, 73bitr4g 317 1 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3135  df-rex 3136 This theorem is referenced by:  r19.23  3300  rexlimd2  3302  riotasv3d  36215
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