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Axiom ax-4 1310
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.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 1310 through ax-7 1256 plus rule ax-gen 1257). 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 1257. Conditional forms of the converse are given by ax-12 1311, ax-15 1686, ax-16 1523, and ax-17 1319.

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 1498.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1326 (which derives ax-4 1310 from certain other axioms) relies on ax-3 7 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.)

Assertion
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 1253 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1311  hbequid  1315  ax5o  1327  ax5  1329  ax6o  1330  ax6  1332  a4i  1335  a4s  1336  a4sd  1337  hbnt  1338  hbim  1342  19.3  1354  19.21  1357  19.21bi  1360  hbimd  1368  19.21t  1370  a6e  1407  19.12  1415  19.38  1420  ax9o  1437  hbae  1457  drex1  1467  equveli  1482  sb2  1490  ax11  1534  ax11b  1535  dfsb2  1540  sbf3t  1564  hbsb4  1565  hbsb4t  1566  sb56  1583  sb6  1584  a16gb  1593  sbal1  1672  mopick  1747  2eu1  1765  ax11indalem  1790  ax11inda2ALT  1791
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