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 )

Proof

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 )