Theorem sb4b

Description: Simplified definition of substitution when variables are distinct. Version of sb6 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

	Ref	Expression
Assertion	sb4b	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nfna1	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑡
2		nfeqf2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
3	1 2	nfan1	⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 )
4		equequ2	⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) )
5	4	imbi1d	⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
6	5	adantl	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
7	3 6	albid	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
8	7	pm5.74da	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
9	8	albidv	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
10		df-sb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
11		ax6ev	⊢ ∃ 𝑦 𝑦 = 𝑡
12	11	a1bi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
13		19.23v	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
14	12 13	bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
15	9 10 14	3bitr4g	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )