Theorem axrep4v 5211

Description: Version of axrep4 5212 with a disjoint variable condition, requiring fewer axioms. (Contributed by Matthew House, 18-Sep-2025.)

Assertion

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

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)

Proof of Theorem axrep4v

Step	Hyp	Ref	Expression
1		ax-rep 5206	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
2		19.3v 1989	. . . . . 6 ⊢ (∀𝑧𝜑 ↔ 𝜑)
3	2	imbi1i 350	. . . . 5 ⊢ ((∀𝑧𝜑 → 𝑦 = 𝑧) ↔ (𝜑 → 𝑦 = 𝑧))
4	3	albii 1826	. . . 4 ⊢ (∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∀𝑦(𝜑 → 𝑦 = 𝑧))
5	4	exbii 1855	. . 3 ⊢ (∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
6	5	albii 1826	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
7	2	anbi2i 629	. . . . . 6 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
8	7	exbii 1855	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
9	8	bibi2i 338	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
10	9	albii 1826	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
11	10	exbii 1855	. 2 ⊢ (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
12	1, 6, 11	3imtr3i 292	1 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  axrep6  5215  axprlem3  5361