Theorem sbequ2

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Wolf Lammen, 3-Feb-2024)

	Ref	Expression
Assertion	sbequ2	⊢ ( 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-sb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2	1	biimpi	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3		equvinva	⊢ ( 𝑥 = 𝑡 → ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) )
4		equcomi	⊢ ( 𝑡 = 𝑦 → 𝑦 = 𝑡 )
5		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
6	4 5	imim12i	⊢ ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑡 = 𝑦 → ( 𝑥 = 𝑦 → 𝜑 ) ) )
7	6	impcomd	⊢ ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
8	7	aleximi	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → ∃ 𝑦 𝜑 ) )
9	2 3 8	syl2im	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ( 𝑥 = 𝑡 → ∃ 𝑦 𝜑 ) )
10		ax5e	⊢ ( ∃ 𝑦 𝜑 → 𝜑 )
11	9 10	syl6com	⊢ ( 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 → 𝜑 ) )