Description: Axiom of Specialization. A universally quantified wff implies the wff without the universal quantifier (i.e., an instance, or special case, of the generalized wff). In other words, if something is true for all x , then it is true for any specific x (that would typically occur as a free variable in the wff substituted for ph ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y , but only x is free in A. y x = y .) Axiom scheme C5' in Megill p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of Tarski p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen . Conditional forms of the converse are given by ax-13 , ax-c14 , ax-c16 , and ax-5 .
Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. In our axiomatization, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution (see stdpc4 ).
An interesting alternate axiomatization uses axc5c711 and ax-c4 in place of ax-c5 , ax-4 , ax-10 , and ax-11 .
This axiom is obsolete and should no longer be used. It is proved above as Theorem sp . (Contributed by NM, 3-Jan-1993) Use sp instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-c5 | ⊢ ( ∀ 𝑥 𝜑 → 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | ⊢ 𝑥 | |
| 1 | wph | ⊢ 𝜑 | |
| 2 | 1 0 | wal | ⊢ ∀ 𝑥 𝜑 |
| 3 | 2 1 | wi | ⊢ ( ∀ 𝑥 𝜑 → 𝜑 ) |