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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	\|- ( A. x ph -> [ t / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		ala1	\|- ( A. x ph -> A. x ( x = y -> ph ) )
2	1	a1d	\|- ( A. x ph -> ( y = t -> A. x ( x = y -> ph ) ) )
3	2	alrimiv	\|- ( A. x ph -> A. y ( y = t -> A. x ( x = y -> ph ) ) )
4		dfsb	\|- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
5	3 4	sylibr	\|- ( A. x ph -> [ t / x ] ph )