Theorem bj-rep 37496

Description: Version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function from ax-rep 5217 (in the form of axrep6 5226). (Contributed by BJ, 14-Mar-2026.) The proof proves the statement without the DV condition on 𝑥, 𝜑, but the DV condition is added to this statement to show that this weaker version is sufficient. (Proof modification is discouraged.) (New usage is discouraged.)

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 3067	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃!𝑧𝜑))
2		eumo 2595	. . . . . . 7 ⊢ (∃!𝑧𝜑 → ∃*𝑧𝜑)
3	2	imim2i 16	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → (𝑦 ∈ 𝑥 → ∃*𝑧𝜑))
4		moanimv 2636	. . . . . 6 ⊢ (∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑) ↔ (𝑦 ∈ 𝑥 → ∃*𝑧𝜑))
5	3, 4	sylibr 236	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → ∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
6	5	alimi 1821	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → ∃!𝑧𝜑) → ∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
7	1, 6	sylbi 219	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑))
8		axrep6 5226	. . . 4 ⊢ (∀𝑦∃*𝑧(𝑦 ∈ 𝑥 ∧ 𝜑) → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)))
9		rexanid 3101	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝑥 𝜑)
10	9	bibi2i 339	. . . . . 6 ⊢ ((𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
11	10	albii 1829	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
12	11	exbii 1858	. . . 4 ⊢ (∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 (𝑦 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
13	8, 12	sylib 220	. . 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 1805	1 ⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1548  ∃wex 1789  ∃*wmo 2554  ∃!weu 2585  ∀wral 3066  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-rep 5217

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-mo 2556  df-eu 2586  df-ral 3067  df-rex 3077

This theorem is referenced by: (None)