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Theorem 19.41 1879
 Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
19.41.1 xψ
Assertion
Ref Expression
19.41 (x(φ ψ) ↔ (xφ ψ))

Proof of Theorem 19.41
StepHypRef Expression
1 19.40 1609 . . 3 (x(φ ψ) → (xφ xψ))
2 19.41.1 . . . . 5 xψ
3 id 19 . . . . 5 (ψψ)
42, 3exlimi 1803 . . . 4 (xψψ)
54anim2i 552 . . 3 ((xφ xψ) → (xφ ψ))
61, 5syl 15 . 2 (x(φ ψ) → (xφ ψ))
7 pm3.21 435 . . . 4 (ψ → (φ → (φ ψ)))
82, 7eximd 1770 . . 3 (ψ → (xφx(φ ψ)))
98impcom 419 . 2 ((xφ ψ) → x(φ ψ))
106, 9impbii 180 1 (x(φ ψ) ↔ (xφ ψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  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-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  19.42  1880  19.41v  1901  eean  1912  r19.41  2763  eliunxp  4821
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