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Theorem spw 1679
 Description: Weak version of specialization scheme sp 1728. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1728 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1728 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1709 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1728 are spfw 1676 (minimal distinct variable requirements), spnfw 1658 (when is not free in ), spvw 1655 (when does not appear in ), sptruw 1684 (when is true), and spfalw 1685 (when is false). (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
spw.1
Assertion
Ref Expression
spw
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem spw
StepHypRef Expression
1 ax-17 1606 . . 3
2 ax-5 1547 . . 3
3 spw.1 . . . . . 6
43biimprd 214 . . . . 5
54equcoms 1666 . . . 4
65spimvw 1657 . . 3
71, 2, 6syl56 30 . 2
83biimpd 198 . . 3
98spimvw 1657 . 2
107, 9mpg 1538 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176  wal 1530 This theorem is referenced by:  spvwOLD  1680  spfalw  1685  hba1w  1693  ax11w  1707 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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