Theorem bdbm1.3ii 10011

Description: Bounded version of bm1.3ii 3878. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdbm1.3ii.bd	⊢ BOUNDED 𝜑
bdbm1.3ii.1	⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)

Assertion

Ref	Expression
bdbm1.3ii	⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Distinct variable groups:   𝜑,𝑥   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bdbm1.3ii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdbm1.3ii.1	. . . . 5 ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)
2		elequ2 1601	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
3	2	imbi2d 219	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑧)))
4	3	albidv 1705	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝜑 → 𝑦 ∈ 𝑧)))
5	4	cbvexv 1795	. . . . 5 ⊢ (∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧))
6	1, 5	mpbi 133	. . . 4 ⊢ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧)
7		bdbm1.3ii.bd	. . . . 5 ⊢ BOUNDED 𝜑
8	7	bdsep1 10005	. . . 4 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))
9	6, 8	pm3.2i 257	. . 3 ⊢ (∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
10	9	exan 1583	. 2 ⊢ ∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
11		19.42v 1786	. . . 4 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) ↔ (∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))))
12		bimsc1 870	. . . . . 6 ⊢ (((𝜑 → 𝑦 ∈ 𝑧) ∧ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → (𝑦 ∈ 𝑥 ↔ 𝜑))
13	12	alanimi 1348	. . . . 5 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
14	13	eximi 1491	. . . 4 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
15	11, 14	sylbir 125	. . 3 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
16	15	exlimiv 1489	. 2 ⊢ (∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
17	10, 16	ax-mp 7	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bj-zfpair2  10030  bj-axun2  10035  bj-uniex2  10036