# 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 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \left[{t}/{x}\right]{\phi }$

### Proof

Step Hyp Ref Expression
1 ala1 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)$
2 1 a1d ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \left({y}={t}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
3 2 alrimiv ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}\left({y}={t}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
4 df-sb ${⊢}\left[{t}/{x}\right]{\phi }↔\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}={t}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
5 3 4 sylibr ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \left[{t}/{x}\right]{\phi }$