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Axiom ax-4 1305
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (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 yx = 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 1243. Conditional forms of the converse are given by ax-12 1307, ax-15 1879, ax-16 1540, and ax-17 1320.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1507.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1877 (which derives ax-4 1305 from certain other axioms) relies on ax-3 714 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.)

Ref Expression
ax-4 (xφφ)

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 set x
31, 2wal 1240 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1306  ax-12  1307  hbequid  1311  a4i  1331  a4s  1332  a4sd  1333  hbim  1339  19.3  1347  19.21  1350  19.21bi  1353  hbimd  1365  19.21ht  1369  hbnt  1414  19.12  1424  19.38  1429  ax9o  1443  hbae  1460  equveli  1488  sb2  1499  drex1  1524  ax11b  1552  sbf3t  1572  hbsb4  1573  hbsb4t  1574  a16gb  1592  sb56  1611  sb6  1612  sbalyz  1704  mopick  1784  2eu1  1802  dfsb2  1860  ax5o  1874  ax5  1876  ax11  1880  ax11indalem  1886  ax11inda2ALT  1887
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