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Theorem spw 1706
 Description: Weak version of specialization scheme sp 1763. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1763 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1763 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1738 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1763 are spfw 1703 (minimal distinct variable requirements), spnfw 1682 (when is not free in ), spvw 1678 (when does not appear in ), sptruw 1683 (when is true), and spfalw 1684 (when is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
Hypothesis
Ref Expression
spw.1
Assertion
Ref Expression
spw
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem spw
StepHypRef Expression
1 ax-17 1626 . 2
2 ax-17 1626 . 2
3 ax-17 1626 . 2
4 spw.1 . 2
51, 2, 3, 4spfw 1703 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177  wal 1549 This theorem is referenced by:  spvwOLD  1708  spfalwOLD  1712  hba1w  1722  ax11w  1736 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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