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Metamath Proof Explorer

Theorem axrep6

Description: A condensed form of ax-rep . (Contributed by SN, 18-Sep-2023) (Proof shortened by Matthew House, 18-Sep-2025)

Ref Expression
Assertion axrep6 ( ∀ 𝑤 ∃* 𝑧 𝜑 → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) )

Proof

Step Hyp Ref Expression
1 axrep4v ⊢ ( ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) )
2 df-mo ⊢ ( ∃* 𝑧 𝜑 ↔ ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) )
3 2 albii ⊢ ( ∀ 𝑤 ∃* 𝑧 𝜑 ↔ ∀ 𝑤 ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) )
4 df-rex ⊢ ( ∃ 𝑤 ∈ 𝑥 𝜑 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) )
5 4 bibi2i ⊢ ( ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) )
6 5 albii ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) )
7 6 exbii ⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑥 ∧ 𝜑 ) ) )
8 1 3 7 3imtr4i ⊢ ( ∀ 𝑤 ∃* 𝑧 𝜑 → ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ∈ 𝑥 𝜑 ) )