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Theorem 19.41 1549
Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
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 1495 . . 3 (x(φ ψ) → (xφ xψ))
2 19.41.1 . . . . 5 xψ
3219.9 1508 . . . 4 (xψψ)
43anbi2i 430 . . 3 ((xφ xψ) ↔ (xφ ψ))
51, 4sylib 127 . 2 (x(φ ψ) → (xφ ψ))
6 pm3.21 251 . . . 4 (ψ → (φ → (φ ψ)))
72, 6eximd 1476 . . 3 (ψ → (xφx(φ ψ)))
87impcom 116 . 2 ((xφ ψ) → x(φ ψ))
95, 8impbii 117 1 (x(φ ψ) ↔ (xφ ψ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wnf 1322  wex 1354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1309  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-4 1373  ax-ial 1400
This theorem depends on definitions:  df-bi 110  df-nf 1323
This theorem is referenced by:  19.42  1551  eean  1779  r19.41  2434  eliunxp  4390  dfopab2  5726  dfoprab3s  5727
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