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		dfmo	⊢ ( ∃* 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 ) )