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Theorem 19.21t 1795
 Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Assertion
Ref Expression
19.21t (Ⅎxφ → (x(φψ) ↔ (φxψ)))

Proof of Theorem 19.21t
StepHypRef Expression
1 nfr 1761 . . 3 (Ⅎxφ → (φxφ))
2 ax-5 1557 . . 3 (x(φψ) → (xφxψ))
31, 2syl9 66 . 2 (Ⅎxφ → (x(φψ) → (φxψ)))
4 19.9t 1779 . . . 4 (Ⅎxφ → (xφφ))
54imbi1d 308 . . 3 (Ⅎxφ → ((xφxψ) ↔ (φxψ)))
6 19.38 1794 . . 3 ((xφxψ) → x(φψ))
75, 6syl6bir 220 . 2 (Ⅎxφ → ((φxψ) → x(φψ)))
83, 7impbid 183 1 (Ⅎxφ → (x(φψ) ↔ (φxψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 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 This theorem is referenced by:  19.21  1796  nfimd  1808  sbcom  2089  sbal2  2134  ax11indalem  2197  ax11inda2ALT  2198  r19.21t  2699  ceqsalt  2881  sbciegft  3076
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