Theorem stdpc5 1630

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 1748. 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	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
stdpc5	⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))

Proof of Theorem stdpc5

Step	Hyp	Ref	Expression
1		stdpc5.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.21 1629	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
3	2	biimpi 120	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1393  Ⅎwnf 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  sbalyz  2050  ra5  3118