Theorem bj-bm1.3ii 35162

Description: The extension of a predicate (𝜑(𝑧)) is included in a set (𝑥) if and only if it is a set (𝑦). Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5218. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.) Generalized to a closed form biconditional with existential quantifications using two different setvars 𝑥, 𝑦 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.)

TODO: move in place of bm1.3ii 5221. Relabel ("sepbi"?).

Assertion

Ref	Expression
bj-bm1.3ii	⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))

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

Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem bj-bm1.3ii

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2123	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑡))
2	1	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑡 → ((𝜑 → 𝑧 ∈ 𝑥) ↔ (𝜑 → 𝑧 ∈ 𝑡)))
3	2	albidv 1924	. . 3 ⊢ (𝑥 = 𝑡 → (∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝜑 → 𝑧 ∈ 𝑡)))
4	3	cbvexvw 2041	. 2 ⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑡∀𝑧(𝜑 → 𝑧 ∈ 𝑡))
5		ax-sep 5218	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))
6		19.42v 1958	. . . . . 6 ⊢ (∃𝑦(∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ∧ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))) ↔ (∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ∧ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))))
7		bimsc1 840	. . . . . . . 8 ⊢ (((𝜑 → 𝑧 ∈ 𝑡) ∧ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))) → (𝑧 ∈ 𝑦 ↔ 𝜑))
8	7	alanimi 1820	. . . . . . 7 ⊢ ((∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ∧ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))) → ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
9	8	eximi 1838	. . . . . 6 ⊢ (∃𝑦(∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ∧ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
10	6, 9	sylbir 234	. . . . 5 ⊢ ((∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ∧ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑡 ∧ 𝜑))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
11	5, 10	mpan2 687	. . . 4 ⊢ (∀𝑧(𝜑 → 𝑧 ∈ 𝑡) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
12	11	exlimiv 1934	. . 3 ⊢ (∃𝑡∀𝑧(𝜑 → 𝑧 ∈ 𝑡) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
13		elequ2 2123	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑡))
14	13	bibi1d 343	. . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑧 ∈ 𝑦 ↔ 𝜑) ↔ (𝑧 ∈ 𝑡 ↔ 𝜑)))
15	14	albidv 1924	. . . . 5 ⊢ (𝑦 = 𝑡 → (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑡 ↔ 𝜑)))
16	15	cbvexvw 2041	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑) ↔ ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ 𝜑))
17		biimpr 219	. . . . . 6 ⊢ ((𝑧 ∈ 𝑡 ↔ 𝜑) → (𝜑 → 𝑧 ∈ 𝑡))
18	17	alimi 1815	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑡 ↔ 𝜑) → ∀𝑧(𝜑 → 𝑧 ∈ 𝑡))
19	18	eximi 1838	. . . 4 ⊢ (∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ 𝜑) → ∃𝑡∀𝑧(𝜑 → 𝑧 ∈ 𝑡))
20	16, 19	sylbi 216	. . 3 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑) → ∃𝑡∀𝑧(𝜑 → 𝑧 ∈ 𝑡))
21	12, 20	impbii 208	. 2 ⊢ (∃𝑡∀𝑧(𝜑 → 𝑧 ∈ 𝑡) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
22	4, 21	bitri 274	1 ⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)