Theorem frege58c

Description: Principle related to sp . Axiom 58 of Frege1879 p. 51. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege58c.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege58c	⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		frege58c.a	⊢ 𝐴 ∈ 𝐵
2		ax-frege58b	⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 )
3		sbsbc	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
4	2 3	sylib	⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 )
5		dfsbcq	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
6	4 5	imbitrid	⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )
7	6	vtocleg	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )
8	1 7	ax-mp	⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 )