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) Revise df-sb again. (Revised by Wolf Lammen, 4-Jun-2026)

	Ref	Expression
Assertion	stdpc4	$$⊢\forall x\phi \to \left[t/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		stdpc4lem	$$⊢\forall x\phi \to \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
2		stdpc4lem	$$⊢\forall x\phi \to \forall z\left(z=t\to \forall x\left(x=z\to \phi \right)\right)$$
3		df-sb	$$⊢\left[t/x\right]\phi \leftrightarrow \left(\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\wedge \forall z\left(z=t\to \forall x\left(x=z\to \phi \right)\right)\right)$$
4	1 2 3	sylanbrc	$$⊢\forall x\phi \to \left[t/x\right]\phi$$