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Theorem stdpc5 2232
 Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 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 2098. 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 2019 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 2174. (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 2231 . 2 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
32biimpi 206 1 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629  Ⅎwnf 1856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858 This theorem is referenced by:  2stdpc5  33152
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