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Theorem spw 1807
Description: Weak version of the specialization scheme sp 1859. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1859 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1859 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1831 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1859 are spfw 1806 (minimal distinct variable requirements), spnfw 1785 (when is not free in ), spvw 1756 (when does not appear in ), sptruw 1630 (when is true), and spfalw 1786 (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:   ,   ,   ,

Proof of Theorem spw
StepHypRef Expression
1 ax-5 1704 . 2
2 ax-5 1704 . 2
3 ax-5 1704 . 2
4 spw.1 . 2
51, 2, 3, 4spfw 1806 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  A.wal 1393
This theorem is referenced by:  hba1w  1814  ax12w  1829  bj-ax12w  34236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790
This theorem depends on definitions:  df-bi 185  df-ex 1613
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