Theorem sbequ1

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993) Revise df-sb . (Revised by BJ, 22-Dec-2020)

	Ref	Expression
Assertion	sbequ1	⊢ ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		equeucl	⊢ ( 𝑥 = 𝑡 → ( 𝑦 = 𝑡 → 𝑥 = 𝑦 ) )
2		ax12v	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	1 2	syl6	⊢ ( 𝑥 = 𝑡 → ( 𝑦 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
4	3	com23	⊢ ( 𝑥 = 𝑡 → ( 𝜑 → ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
5	4	alrimdv	⊢ ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
6		df-sb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
7	5 6	imbitrrdi	⊢ ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) )