New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  19.21tOLD GIF version

Theorem 19.21tOLD 1863
 Description: Obsolete proof of 19.21t 1795 as of 30-Dec-2017. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
Assertion
Ref Expression
19.21tOLD (Ⅎxφ → (x(φψ) ↔ (φxψ)))

Proof of Theorem 19.21tOLD
StepHypRef Expression
1 id 19 . . . 4 (Ⅎxφ → Ⅎxφ)
21nfrd 1763 . . 3 (Ⅎxφ → (φxφ))
3 alim 1558 . . 3 (x(φψ) → (xφxψ))
42, 3syl9 66 . 2 (Ⅎxφ → (x(φψ) → (φxψ)))
5 nfa1 1788 . . . . . 6 xxψ
65a1i 10 . . . . 5 (Ⅎxφ → Ⅎxxψ)
71, 6nfimd 1808 . . . 4 (Ⅎxφ → Ⅎx(φxψ))
87nfrd 1763 . . 3 (Ⅎxφ → ((φxψ) → x(φxψ)))
9 sp 1747 . . . . 5 (xψψ)
109imim2i 13 . . . 4 ((φxψ) → (φψ))
1110alimi 1559 . . 3 (x(φxψ) → x(φψ))
128, 11syl6 29 . 2 (Ⅎxφ → ((φxψ) → x(φψ)))
134, 12impbid 183 1 (Ⅎxφ → (x(φψ) ↔ (φxψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  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: (None)
 Copyright terms: Public domain W3C validator