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Theorem stdpc5 1798
Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis xφ can be thought of as emulating "x is not free in φ." 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 1678. 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.)
Ref Expression
stdpc5.1 xφ
Ref Expression
stdpc5 (x(φψ) → (φxψ))

Proof of Theorem stdpc5
StepHypRef Expression
1 stdpc5.1 . . 3 xφ
2119.21 1796 . 2 (x(φψ) ↔ (φxψ))
32biimpi 186 1 (x(φψ) → (φxψ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wnf 1544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545
This theorem is referenced by:  ra5  3130
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