Theorem bj-axrep3 33620

Description: Remove dependency on ax-13 2301 from axrep3 5054. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axrep3	⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

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

Proof of Theorem bj-axrep3

Step	Hyp	Ref	Expression
1		nfe1 2087	. . . 4 ⊢ Ⅎ𝑦∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦)
2		nfv 1873	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑥
3		nfv 1873	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑤
4		nfa1 2088	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦𝜑
5	3, 4	nfan 1862	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
6	5	nfex 2264	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
7	2, 6	nfbi 1866	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
8	7	nfal 2263	. . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
9	1, 8	nfim 1859	. . 3 ⊢ Ⅎ𝑦(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
10	9	nfex 2264	. 2 ⊢ Ⅎ𝑦∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
11		elequ2 2064	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑤))
12	11	anbi1d 620	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
13	12	exbidv 1880	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
14	13	bibi2d 335	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))))
15	14	albidv 1879	. . . 4 ⊢ (𝑦 = 𝑤 → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))))
16	15	imbi2d 333	. . 3 ⊢ (𝑦 = 𝑤 → ((∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))))
17	16	exbidv 1880	. 2 ⊢ (𝑦 = 𝑤 → (∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))))
18		bj-axrep2 33619	. 2 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
19	10, 17, 18	bj-chvarv 33573	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀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 5050

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-axrep4  33621