Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.21t Structured version   Visualization version   GIF version

Theorem r19.21t 2950
 Description: Restricted quantifier version of 19.21t 2071; closed form of r19.21 2951. (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 2071 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓))))
2 df-ral 2912 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 bi2.04 376 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
43albii 1744 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
52, 4bitri 264 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
6 df-ral 2912 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
76imbi2i 326 . 2 ((𝜑 → ∀𝑥𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
81, 5, 73bitr4g 303 1 (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707  df-ral 2912 This theorem is referenced by:  r19.21  2951
 Copyright terms: Public domain W3C validator