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Theorem stdpc5 1476
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 1590. 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 𝑥𝜑
Assertion
Ref Expression
stdpc5 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Proof of Theorem stdpc5
StepHypRef Expression
1 stdpc5.1 . . 3 𝑥𝜑
2119.21 1475 . 2 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
32biimpi 113 1 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241  wnf 1349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by:  sbalyz  1875  ra5  2846
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