Theorem ralsng

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝜑 ) )
3		velsn	⊢ ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 )
4	3	imbi1i	⊢ ( ( 𝑥 ∈ { 𝐴 } → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜑 ) )
5	4	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )
6	2 5	bitri	⊢ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )
7		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
8	1	pm5.74i	⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜓 ) )
9	8	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) )
10	9	a1i	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
11		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
12	11	a1i	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
13		pm5.5	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜓 ) )
14	10 12 13	3bitrd	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
15	7 14	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
16	6 15	syl5bb	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )