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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	$⊢ \forall x φ \to [t/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		ala1	$⊢ \forall x φ \to \forall x (x = y \to φ)$
2	1	a1d	$⊢ \forall x φ \to (y = t \to \forall x (x = y \to φ))$
3	2	alrimiv	$⊢ \forall x φ \to \forall y (y = t \to \forall x (x = y \to φ))$
4		dfsb	$⊢ [t/ x] φ \leftrightarrow \forall y (y = t \to \forall x (x = y \to φ))$
5	3 4	sylibr	$⊢ \forall x φ \to [t/ x] φ$