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Theorem 19.23t 1564
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 1487 . . 3 (x(φψ) → (xφxψ))
2 19.9t 1530 . . . 4 (Ⅎxψ → (xψψ))
32biimpd 132 . . 3 (Ⅎxψ → (xψψ))
41, 3syl9r 67 . 2 (Ⅎxψ → (x(φψ) → (xφψ)))
5 nfr 1408 . . . 4 (Ⅎxψ → (ψxψ))
65imim2d 48 . . 3 (Ⅎxψ → ((xφψ) → (xφxψ)))
7 19.38 1563 . . 3 ((xφxψ) → x(φψ))
86, 7syl6 29 . 2 (Ⅎxψ → ((xφψ) → x(φψ)))
94, 8impbid 120 1 (Ⅎxψ → (x(φψ) ↔ (xφψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wnf 1346  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  19.23  1565  r19.23t  2417  ceqsalt  2574  vtoclgft  2598  sbciegft  2787
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