Theorem axrep4v

Description: Version of axrep4 with a disjoint variable condition, requiring fewer axioms. (Contributed by Matthew House, 18-Sep-2025)

	Ref	Expression
Assertion	axrep4v	⊢ ( ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) → ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-rep	⊢ ( ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( ∀ 𝑧 𝜑 → 𝑦 = 𝑧 ) → ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ) )
2		19.3v	⊢ ( ∀ 𝑧 𝜑 ↔ 𝜑 )
3	2	imbi1i	⊢ ( ( ∀ 𝑧 𝜑 → 𝑦 = 𝑧 ) ↔ ( 𝜑 → 𝑦 = 𝑧 ) )
4	3	albii	⊢ ( ∀ 𝑦 ( ∀ 𝑧 𝜑 → 𝑦 = 𝑧 ) ↔ ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) )
5	4	exbii	⊢ ( ∃ 𝑧 ∀ 𝑦 ( ∀ 𝑧 𝜑 → 𝑦 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) )
6	5	albii	⊢ ( ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( ∀ 𝑧 𝜑 → 𝑦 = 𝑧 ) ↔ ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) )
7	2	anbi2i	⊢ ( ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ↔ ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) )
9	8	bibi2i	⊢ ( ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) ) )
10	9	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) ) )
11	10	exbii	⊢ ( ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) ) )
12	1 6 11	3imtr3i	⊢ ( ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) → ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ 𝜑 ) ) )