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Theorem r19.21t 3135
Description: Restricted quantifier version of 19.21t 2204; closed form of r19.21 3136. (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 2204 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓))))
2 df-ral 3066 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 bi2.04 392 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
43albii 1827 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
52, 4bitri 278 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
6 df-ral 3066 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
76imbi2i 339 . 2 ((𝜑 → ∀𝑥𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
81, 5, 73bitr4g 317 1 (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541  wnf 1791  wcel 2110  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792  df-ral 3066
This theorem is referenced by:  r19.21  3136
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