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Theorem r19.21t 2699
 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 → (x A (φψ) ↔ (φx A ψ)))

Proof of Theorem r19.21t
StepHypRef Expression
1 bi2.04 350 . . . 4 ((x A → (φψ)) ↔ (φ → (x Aψ)))
21albii 1566 . . 3 (x(x A → (φψ)) ↔ x(φ → (x Aψ)))
3 19.21t 1795 . . 3 (Ⅎxφ → (x(φ → (x Aψ)) ↔ (φx(x Aψ))))
42, 3syl5bb 248 . 2 (Ⅎxφ → (x(x A → (φψ)) ↔ (φx(x Aψ))))
5 df-ral 2619 . 2 (x A (φψ) ↔ x(x A → (φψ)))
6 df-ral 2619 . . 3 (x A ψx(x Aψ))
76imbi2i 303 . 2 ((φx A ψ) ↔ (φx(x Aψ)))
84, 5, 73bitr4g 279 1 (Ⅎxφ → (x A (φψ) ↔ (φx A ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  r19.21  2700
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