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Theorem stdpc5 2175
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 2001. 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 1920 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 2114. (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 2174 . 2 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
32biimpi 217 1 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1523  wnf 1769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-12 2143
This theorem depends on definitions:  df-bi 208  df-ex 1766  df-nf 1770
This theorem is referenced by:  2stdpc5  33728
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