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Theorem stdpc5 1492
Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis  F/ x ph can be thought of as emulating " x is not free in  ph." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example  x would not (for us) be free in  x  =  x by nfequid 1606. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypothesis
Ref Expression
stdpc5.1  |-  F/ x ph
Assertion
Ref Expression
stdpc5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
)

Proof of Theorem stdpc5
StepHypRef Expression
1 stdpc5.1 . . 3  |-  F/ x ph
2119.21 1491 . 2  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
32biimpi 117 1  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   F/wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  sbalyz  1891  ra5  2874
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