Theorem sbequ2OLD

Description: Obsolete version of sbequ2 as of 3-Feb-2024. (Contributed by NM, 16-May-1993) (Proof shortened by Wolf Lammen, 25-Feb-2018) Revise df-sb . (Revised by BJ, 22-Dec-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sbequ2OLD	⊢ ( 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		equvinva	⊢ ( 𝑥 = 𝑡 → ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) )
2		df-sb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3		equcomi	⊢ ( 𝑡 = 𝑦 → 𝑦 = 𝑡 )
4		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
5	3 4	imim12i	⊢ ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑡 = 𝑦 → ( 𝑥 = 𝑦 → 𝜑 ) ) )
6	5	impcomd	⊢ ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
7	6	alimi	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ( ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
8	2 7	sylbi	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ∀ 𝑦 ( ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
9		19.23v	⊢ ( ∀ 𝑦 ( ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) ↔ ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
10	8 9	sylib	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝑡 = 𝑦 ) → 𝜑 ) )
11	1 10	syl5com	⊢ ( 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 → 𝜑 ) )