Theorem bj-axrep2 33619

Description: Remove dependency on ax-13 2301 from axrep2 5046. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axrep2	⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem bj-axrep2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfe1 2087	. . . . 5 ⊢ Ⅎ𝑤∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
2		nfv 1873	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
3	1, 2	nfim 1859	. . . 4 ⊢ Ⅎ𝑤(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
4	3	nfex 2264	. . 3 ⊢ Ⅎ𝑤∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
5		elequ2 2064	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
6	5	anbi1d 620	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
7	6	exbidv 1880	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
8	7	bibi2d 335	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
9	8	albidv 1879	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
10	9	imbi2d 333	. . . 4 ⊢ (𝑤 = 𝑦 → ((∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) ↔ (∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))))
11	10	exbidv 1880	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))))
12		bj-axrep1 33618	. . 3 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
13	4, 11, 12	bj-chvarv 33573	. 2 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
14		sp 2111	. . . . . . 7 ⊢ (∀𝑦𝜑 → 𝜑)
15	14	imim1i 63	. . . . . 6 ⊢ ((𝜑 → 𝑧 = 𝑦) → (∀𝑦𝜑 → 𝑧 = 𝑦))
16	15	alimi 1774	. . . . 5 ⊢ (∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
17	16	eximi 1797	. . . 4 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
18		nfv 1873	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
19		nfa1 2088	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦𝜑
20		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
21	19, 20	nfim 1859	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦𝜑 → 𝑧 = 𝑤)
22	21	nfal 2263	. . . . 5 ⊢ Ⅎ𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
23		equequ2 1983	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 = 𝑦 ↔ 𝑧 = 𝑤))
24	23	imbi2d 333	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (∀𝑦𝜑 → 𝑧 = 𝑤)))
25	24	albidv 1879	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)))
26	18, 22, 25	cbvexv1 2278	. . . 4 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
27	17, 26	sylib 210	. . 3 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
28	27	imim1i 63	. 2 ⊢ ((∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) → (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
29	13, 28	eximii 1799	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-rep 5043

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747

This theorem is referenced by:  bj-axrep3  33620