Theorem axrep4 5290

Description: A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Matthew House, 18-Sep-2025.)

Hypothesis

Ref	Expression
axrep4.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
axrep4	⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

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

Proof of Theorem axrep4

Step	Hyp	Ref	Expression
1		ax-rep 5284	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
2		axrep4.1	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
3	2	19.3 2199	. . . . . 6 ⊢ (∀𝑧𝜑 ↔ 𝜑)
4	3	imbi1i 349	. . . . 5 ⊢ ((∀𝑧𝜑 → 𝑦 = 𝑧) ↔ (𝜑 → 𝑦 = 𝑧))
5	4	albii 1815	. . . 4 ⊢ (∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∀𝑦(𝜑 → 𝑦 = 𝑧))
6	5	exbii 1844	. . 3 ⊢ (∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
7	6	albii 1815	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(∀𝑧𝜑 → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
8	3	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
9	8	exbii 1844	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
10	9	bibi2i 337	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
11	10	albii 1815	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
12	11	exbii 1844	. 2 ⊢ (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
13	1, 7, 12	3imtr3i 291	1 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1534  ∃wex 1775  Ⅎwnf 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174  ax-rep 5284

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-nf 1780

This theorem is referenced by:  axrep5  5292  axprlem3OLD  5433  funimaexgOLD  6654