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Theorem 19.35 1600
 Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.35 (x(φψ) ↔ (xφxψ))

Proof of Theorem 19.35
StepHypRef Expression
1 19.26 1593 . . . 4 (x(φ ¬ ψ) ↔ (xφ x ¬ ψ))
2 annim 414 . . . . 5 ((φ ¬ ψ) ↔ ¬ (φψ))
32albii 1566 . . . 4 (x(φ ¬ ψ) ↔ x ¬ (φψ))
4 alnex 1543 . . . . 5 (x ¬ ψ ↔ ¬ xψ)
54anbi2i 675 . . . 4 ((xφ x ¬ ψ) ↔ (xφ ¬ xψ))
61, 3, 53bitr3i 266 . . 3 (x ¬ (φψ) ↔ (xφ ¬ xψ))
7 alnex 1543 . . 3 (x ¬ (φψ) ↔ ¬ x(φψ))
8 annim 414 . . 3 ((xφ ¬ xψ) ↔ ¬ (xφxψ))
96, 7, 83bitr3i 266 . 2 x(φψ) ↔ ¬ (xφxψ))
109con4bii 288 1 (x(φψ) ↔ (xφxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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-an 360  df-ex 1542 This theorem is referenced by:  19.35i  1601  19.35ri  1602  19.25  1603  19.43  1605  speimfw  1645  19.39  1661  19.24  1662  19.36  1871  19.37  1873  sbequi  2059
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