Theorem axrep2 5193

Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.) Remove dependency on ax-13 2390. (Revised by BJ, 31-May-2019.)

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 2154	. . . . 5 ⊢ Ⅎ𝑤∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
2		nfv 1915	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
3	1, 2	nfim 1897	. . . 4 ⊢ Ⅎ𝑤(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
4	3	nfex 2343	. . 3 ⊢ Ⅎ𝑤∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
5		axreplem 5192	. . 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)))))
6		axrep1 5191	. . 3 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
7	4, 5, 6	chvarfv 2242	. 2 ⊢ ∃𝑥(∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
8		sp 2182	. . . . . . 7 ⊢ (∀𝑦𝜑 → 𝜑)
9	8	imim1i 63	. . . . . 6 ⊢ ((𝜑 → 𝑧 = 𝑦) → (∀𝑦𝜑 → 𝑧 = 𝑦))
10	9	alimi 1812	. . . . 5 ⊢ (∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
11	10	eximi 1835	. . . 4 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
12		nfv 1915	. . . . 5 ⊢ Ⅎ𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
13		nfa1 2155	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦𝜑
14		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
15	13, 14	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦𝜑 → 𝑧 = 𝑤)
16	15	nfal 2342	. . . . 5 ⊢ Ⅎ𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)
17		equequ2 2033	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 = 𝑦 ↔ 𝑧 = 𝑤))
18	17	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (∀𝑦𝜑 → 𝑧 = 𝑤)))
19	18	albidv 1921	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤)))
20	12, 16, 19	cbvexv1 2362	. . . 4 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
21	11, 20	sylib 220	. . 3 ⊢ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑤∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑤))
22	21	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))))
23	7, 22	eximii 1837	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-rep 5190

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  axrep3  5194  axrepndlem1  10014