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Theorem r19.21t 3250
Description: Restricted quantifier version of 19.21t 2199; closed form of r19.21 3251. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
Assertion
Ref Expression
r19.21t (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Proof of Theorem r19.21t
StepHypRef Expression
1 19.21t 2199 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓))))
2 df-ral 3062 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 bi2.04 388 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
43albii 1821 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
52, 4bitri 274 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
6 df-ral 3062 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
76imbi2i 335 . 2 ((𝜑 → ∀𝑥𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
81, 5, 73bitr4g 313 1 (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wnf 1785  wcel 2106  wral 3061
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 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-ex 1782  df-nf 1786  df-ral 3062
This theorem is referenced by:  r19.21  3251
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