Theorem zfrep6 5224

Description: A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5231 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 5212. (Contributed by NM, 10-Oct-2003.) Shorten proof and reduce axiom dependencies. (Revised by BJ, 5-Apr-2026.)

Assertion

Ref	Expression
zfrep6	⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

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

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

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		euex 2578	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 3075	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
3		df-ral 3053	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
4		eumo 2579	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
5	4	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
6		moanimv 2620	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
7	5, 6	sylibr 234	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
8	7	alimi 1813	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
9	3, 8	sylbi 217	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
10		axrep6 5221	. . . 4 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)))
11		rexanid 3087	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑧 𝜑)
12	11	bibi2i 337	. . . . . 6 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
13	12	albii 1821	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
14	13	exbii 1850	. . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
15	10, 14	sylib 218	. . 3 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
16	9, 15	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
17		replem 5223	. 2 ⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
18	2, 16, 17	syl2anc 585	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781  ∃*wmo 2538  ∃!weu 2569  ∀wral 3052  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-11 2163  ax-rep 5212

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2540  df-eu 2570  df-ral 3053  df-rex 3063

This theorem is referenced by:  bnj865  35081