Metamath Proof Explorer

Theorem stdpc4

Description: The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x , then it also holds for the specific case of t (properly) substituted for x . Translated to traditional notation, it can be read: " A. x ph ( x ) -> ph ( t ) , provided that t is free for x in ph ( x ) ". Axiom 4 of Mendelson p. 69. See also spsbc and rspsbc . (Contributed by NM, 14-May-1993) Revise df-sb . (Revised by BJ, 22-Dec-2020)

Ref Expression
Assertion stdpc4 x φ t x φ


Step Hyp Ref Expression
1 ala1 x φ x x = y φ
2 1 a1d x φ y = t x x = y φ
3 2 alrimiv x φ y y = t x x = y φ
4 df-sb t x φ y y = t x x = y φ
5 3 4 sylibr x φ t x φ