Theorem sb4bOLD

Description: Obsolete version of sb4b as of 21-Feb-2024. (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 25-Jul-2023) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sb4bOLD	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nfeqf2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
2		nfnf1	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝑦 = 𝑡
3		id	⊢ ( Ⅎ 𝑥 𝑦 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
4	2 3	nfan1	⊢ Ⅎ 𝑥 ( Ⅎ 𝑥 𝑦 = 𝑡 ∧ 𝑦 = 𝑡 )
5		equequ2	⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) )
6	5	imbi1d	⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
7	6	adantl	⊢ ( ( Ⅎ 𝑥 𝑦 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
8	4 7	albid	⊢ ( ( Ⅎ 𝑥 𝑦 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
9	8	pm5.74da	⊢ ( Ⅎ 𝑥 𝑦 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
10	1 9	syl	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
11	10	albidv	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
12		df-sb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
13		ax6ev	⊢ ∃ 𝑦 𝑦 = 𝑡
14	13	a1bi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
15		19.23v	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
16	14 15	bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
17	11 12 16	3bitr4g	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )