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Axiom ax-4 1333
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 1269. Conditional forms of the converse are given by ax-12 1335, ax-15 1917, ax-16 1570, and ax-17 1349.

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 1537.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1915 (which derives ax-4 1333 from certain other axioms) relies on ax-3 719 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 1266 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1334  ax-12  1335  hbequid  1339  a4i  1360  a4s  1361  a4sd  1362  hbim  1368  19.3  1376  19.21  1379  19.21bi  1382  hbimd  1394  19.21ht  1399  hbnt  1444  19.12  1455  19.38  1460  ax9o  1474  hbae  1491  equveli  1518  sb2  1529  drex1  1554  ax11b  1582  sbf3t  1603  a16gb  1617  sb56  1636  sb6  1637  sbalyz  1737  hbsb4t  1749  mopick  1823  2eu1  1841  dfsb2  1897  ax5o  1912  ax5  1914  ax11  1918  ax11indalem  1924  ax11inda2ALT  1925
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