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Theorem 19.43 1605
 Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.43 (x(φ ψ) ↔ (xφ xψ))

Proof of Theorem 19.43
StepHypRef Expression
1 df-or 359 . . . 4 ((φ ψ) ↔ (¬ φψ))
21exbii 1582 . . 3 (x(φ ψ) ↔ xφψ))
3 19.35 1600 . . 3 (xφψ) ↔ (x ¬ φxψ))
4 alnex 1543 . . . 4 (x ¬ φ ↔ ¬ xφ)
54imbi1i 315 . . 3 ((x ¬ φxψ) ↔ (¬ xφxψ))
62, 3, 53bitri 262 . 2 (x(φ ψ) ↔ (¬ xφxψ))
7 df-or 359 . 2 ((xφ xψ) ↔ (¬ xφxψ))
86, 7bitr4i 243 1 (x(φ ψ) ↔ (xφ xψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1542 This theorem is referenced by:  19.34  1663  19.44  1877  19.45  1878  rexun  3443  unipr  3905  uniun  3910  opeq  4619  unopab  4638  dmun  4912  coundi  5082  coundir  5083
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