Theorem bj-rep 37312

Description: Version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function from ax-rep 5226 (in the form of axrep6 5235). (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraded.)

Assertion

Ref	Expression
bj-rep	⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))

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

Allowed substitution hints:   𝜑(𝑦,𝑧)

Proof of Theorem bj-rep

Step	Hyp	Ref	Expression
1		df-ral 3053	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃!𝑧𝜑))
2		eumo 2579	. . . . . . 7 ⊢ (∃!𝑧𝜑 → ∃*𝑧𝜑)
3	2	imim2i 16	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → (𝑦 ∈ 𝑥 → ∃*𝑧𝜑))
4		moanimv 2620	. . . . . 6 ⊢ (∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑) ↔ (𝑦 ∈ 𝑥 → ∃*𝑧𝜑))
5	3, 4	sylibr 234	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → ∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
6	5	alimi 1813	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → ∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
7	1, 6	sylbi 217	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
8		axrep6 5235	. . . 4 ⊢ (∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑) → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)))
9		rexanid 3087	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝑥 𝜑)
10	9	bibi2i 337	. . . . . 6 ⊢ ((𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
11	10	albii 1821	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
12	11	exbii 1850	. . . 4 ⊢ (∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
13	8, 12	sylib 218	. . 3 ⊢ (∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑) → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
14	7, 13	syl 17	. 2 ⊢ (∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
15	14	ax-gen 1797	1 ⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 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-rep 5226

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: (None)