Theorem stdpc5 2206

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 2021. 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 1946 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 2141. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)

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 2205	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
3	2	biimpi 219	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1541  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792

This theorem is referenced by:  2stdpc5  34749