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Metamath Proof Explorer

Theorem ralrab2

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015)

Ref Expression
Hypothesis ralab2.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
Assertion ralrab2 ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )

Proof

Step Hyp Ref Expression
1 ralab2.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
2 df-rab ⊢ { 𝑦 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) }
3 2 raleqi ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) } 𝜓 )
4 1 ralab2 ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) } 𝜓 ↔ ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) )
5 impexp ⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
6 5 albii ⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
7 df-ral ⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
8 6 7 bitr4i ⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )
9 3 4 8 3bitri ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )