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Theorem stdpc5 2079
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 1942. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1869 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2021. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
Hypothesis
Ref Expression
stdpc5.1 𝑥𝜑
Assertion
Ref Expression
stdpc5 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Proof of Theorem stdpc5
StepHypRef Expression
1 stdpc5.1 . . 3 𝑥𝜑
2119.21 2078 . 2 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
32biimpi 206 1 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  2stdpc5  32432
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