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Axiom ax-4 1271
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 , it is true for any specific (that would typically occur as a free variable in the wff substituted for ). (A free variable is one that does not occur in the scope of a quantifier: and are both free in , but only is free in .) This is one of the axioms of what we call "pure" predicate calculus (ax-4 1271 through ax-7 1217 plus rule ax-gen 1218). 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 1218. Conditional forms of the converse are given by ax-12 1272, ax-15 1660, ax-16 1481, and ax-17 1280.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 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 1451.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1808 (which derives ax-4 1271 from certain other axioms) relies on ax-3 714 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-4

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2wal 1214 . 2
43, 1wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1272  hbequid  1276  ax5o  1287  ax5  1289  ax6o  1290  ax6  1292  a4i  1295  a4s  1296  a4sd  1297  hbnt  1298  hbim  1302  19.3  1313  19.21  1316  19.21bi  1319  hbimd  1325  19.21t  1327  a6e  1364  19.12  1372  19.38  1375  ax9o  1390  hbae  1408  drex1  1420  equveli  1435  sb2  1443  ax11  1492  ax11b  1493  dfsb2  1498  sbf3t  1524  hbsb4  1525  hbsb4t  1526  a16gb  1549  sb56  1571  sb6  1572  sbal1  1644  sbalyz  1646  mopick  1726  2eu1  1744  ax11indalem  1814  ax11inda2ALT  1815
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