Theorem stdpc5 739

Description: An axiom of standard predicate calculus. Axiom 5 of [Mendelson] p. 59. The hypothesis (φ → ∀xφ) can be thought of as "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 since (by eqid 810 and hbth 697) we can prove (x = x → ∀xx = x).

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	19.21 738	. 2 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))
3	2	biimp 133	1 ⊢ (∀x(φ → ψ) → (φ → ∀xψ))

Syntax hints:   → wi 2  ∀wal 672

This theorem is referenced by:  rax5 1472

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128

metamath.org