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Metamath Proof Explorer

Theorem r19.23t

Description: Closed theorem form of r19.23 . (Contributed by NM, 4-Mar-2013) (Revised by Mario Carneiro, 8-Oct-2016)

Ref Expression
Assertion r19.23t ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 19.23t ⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
3 impexp ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
4 3 albii ⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
5 2 4 bitr4i ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
6 df-rex ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
7 6 imbi1i ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
8 1 5 7 3bitr4g ⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )