Theorem zfrep6 5218

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 5225 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 5206. (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 2581	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 3077	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
3		df-ral 3055	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
4		eumo 2582	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
5	4	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
6		moanimv 2623	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
7	5, 6	sylibr 235	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
8	7	alimi 1818	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
9	3, 8	sylbi 218	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
10		axrep6 5215	. . . 4 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)))
11		rexanid 3089	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑧 𝜑)
12	11	bibi2i 338	. . . . . 6 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
13	12	albii 1826	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
14	13	exbii 1855	. . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑))
15	10, 14	sylib 219	. . 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 5217	. 2 ⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
18	2, 16, 17	syl2anc 590	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786  ∃*wmo 2541  ∃!weu 2572  ∀wral 3054  ∃wrex 3064

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-7 2015  ax-11 2168  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-ral 3055  df-rex 3065

This theorem is referenced by:  bnj865  35112