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Theorem r19.21t 2437
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
Assertion
Ref Expression
r19.21t (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))

Proof of Theorem r19.21t
StepHypRef Expression
1 bi2.04 246 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
21albii 1400 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
3 19.21t 1515 . . 3 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓))))
42, 3syl5bb 190 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓))))
5 df-ral 2354 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
6 df-ral 2354 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
76imbi2i 224 . 2 ((𝜑 → ∀𝑥𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
84, 5, 73bitr4g 221 1 (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wnf 1390  wcel 1434  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354
This theorem is referenced by:  r19.21  2438
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