Theorem stdpc5 1060

Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis (φ → ∀xφ) can be thought of as emulating "x is not free in φ." With this convention, 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 hbequid 1171. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5.

Hypothesis

Ref	Expression
stdpc5.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem stdpc5

Step	Hyp	Ref	Expression
1		stdpc5.1	. . 3 ⊢ (φ → ∀xφ)
2	1	19.21 1058	. 2 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))
3	2	biimp 151	1 ⊢ (∀x(φ → ψ) → (φ → ∀xψ))

Syntax hints:   → wi 3  ∀wal 956

This theorem is referenced by:  ra5 2003

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147

Copyright terms: Public domain