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Metamath Proof Explorer

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 GG, 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 bitrid ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )