ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.21ht Structured version   GIF version

Theorem 19.21ht 1470
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
Assertion
Ref Expression
19.21ht (x(φxφ) → (x(φψ) ↔ (φxψ)))

Proof of Theorem 19.21ht
StepHypRef Expression
1 alim 1343 . . . . 5 (x(φψ) → (xφxψ))
21imim2d 48 . . . 4 (x(φψ) → ((φxφ) → (φxψ)))
32com12 27 . . 3 ((φxφ) → (x(φψ) → (φxψ)))
43sps 1427 . 2 (x(φxφ) → (x(φψ) → (φxψ)))
5 hba1 1430 . . . 4 (x(φxφ) → xx(φxφ))
6 ax-4 1397 . . . 4 (x(φxφ) → (φxφ))
7 hba1 1430 . . . . 5 (xψxxψ)
87a1i 9 . . . 4 (x(φxφ) → (xψxxψ))
95, 6, 8hbimd 1462 . . 3 (x(φxφ) → ((φxψ) → x(φxψ)))
10 ax-4 1397 . . . . 5 (xψψ)
1110imim2i 12 . . . 4 ((φxψ) → (φψ))
1211alimi 1341 . . 3 (x(φxψ) → x(φψ))
139, 12syl6 29 . 2 (x(φxφ) → ((φxψ) → x(φψ)))
144, 13impbid 120 1 (x(φxφ) → (x(φψ) ↔ (φxψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  19.21t  1471  sbal2  1895
  Copyright terms: Public domain W3C validator