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Theorem 19.23t 1800
 Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Assertion
Ref Expression
19.23t (Ⅎxψ → (x(φψ) ↔ (xφψ)))

Proof of Theorem 19.23t
StepHypRef Expression
1 exim 1575 . . 3 (x(φψ) → (xφxψ))
2 19.9t 1779 . . . 4 (Ⅎxψ → (xψψ))
32biimpd 198 . . 3 (Ⅎxψ → (xψψ))
41, 3syl9r 67 . 2 (Ⅎxψ → (x(φψ) → (xφψ)))
5 nfr 1761 . . . 4 (Ⅎxψ → (ψxψ))
65imim2d 48 . . 3 (Ⅎxψ → ((xφψ) → (xφxψ)))
7 19.38 1794 . . 3 ((xφxψ) → x(φψ))
86, 7syl6 29 . 2 (Ⅎxψ → ((xφψ) → x(φψ)))
94, 8impbid 183 1 (Ⅎxψ → (x(φψ) ↔ (xφψ)))
 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  19.23  1801  sbft  2025  axie2  2329  r19.23t  2728  ceqsalt  2881  vtoclgft  2905  sbciegft  3076
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