Theorem axrep2 4701

Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.)

Assertion

Ref	Expression
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 axrep2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfe1 2014	. . . . 5 ⊢ Ⅎ𝑤∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
2		nfv 1830	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
3	1, 2	nfim 1813	. . . 4 ⊢ Ⅎ𝑤(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
4	3	nfex 2140	. . 3 ⊢ Ⅎ𝑤∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
5		elequ2 1991	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
6	5	anbi1d 737	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
7	6	exbidv 1837	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
8	7	bibi2d 331	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
9	8	albidv 1836	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
10	9	imbi2d 329	. . . 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 1837	. . 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		axrep1 4700	. . 3 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
13	4, 11, 12	chvar 2250	. 2 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
14		sp 2041	. . . . . . 7 ⊢ (∀𝑦𝜑 → 𝜑)
15	14	imim1i 61	. . . . . 6 ⊢ ((𝜑 → 𝑧 = 𝑦) → (∀𝑦𝜑 → 𝑧 = 𝑦))
16	15	alimi 1730	. . . . 5 ⊢ (∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
17	16	eximi 1752	. . . 4 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
18		nfv 1830	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
19		nfa1 2015	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦𝜑
20		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
21	19, 20	nfim 1813	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦𝜑 → 𝑧 = 𝑤)
22	21	nfal 2139	. . . . 5 ⊢ Ⅎ𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
23		equequ2 1940	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 = 𝑦 ↔ 𝑧 = 𝑤))
24	23	imbi2d 329	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (∀𝑦𝜑 → 𝑧 = 𝑤)))
25	24	albidv 1836	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)))
26	18, 22, 25	cbvex 2260	. . . 4 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
27	17, 26	sylib 207	. . 3 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
28	27	imim1i 61	. 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 1754	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-rep 4699

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  axrep3  4702  axrepndlem1  9293