Theorem axrep1 5232

Description: The version of the Axiom of Replacement used in the Metamath Solitaire applet https://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 5231 → axrep1 5232 → axrep2 5234 → axrepnd 10683 → zfcndrep 10703 = ax-rep 5231. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2401. (Revised by BJ, 31-May-2019.)

Assertion

Ref	Expression
axrep1	⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑧)

Proof of Theorem axrep1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2159	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
2	1	anbi1d 643	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)))
3	2	exbidv 1953	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
4	3	bibi2d 345	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
5	4	albidv 1952	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
6	5	exbidv 1953	. . . 4 ⊢ (𝑤 = 𝑦 → (∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
7	6	imbi2d 343	. . 3 ⊢ (𝑤 = 𝑦 → ((∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))) ↔ (∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))))
8		ax-rep 5231	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
9		19.3v 2014	. . . . . . . 8 ⊢ (∀𝑦𝜑 ↔ 𝜑)
10	9	imbi1i 352	. . . . . . 7 ⊢ ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (𝜑 → 𝑧 = 𝑦))
11	10	albii 1851	. . . . . 6 ⊢ (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(𝜑 → 𝑧 = 𝑦))
12	11	exbii 1880	. . . . 5 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
13	12	albii 1851	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
14		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑦
15		nfe1 2186	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
16	14, 15	nfbi 1935	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
17	16	nfal 2353	. . . . 5 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
18		nfv 1946	. . . . 5 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
19		elequ2 2159	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥))
20	9	anbi2i 635	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
21	20	exbii 1880	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
22	21	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
23	19, 22	bibi12d 348	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
24	23	albidv 1952	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
25	17, 18, 24	cbvexv1 2371	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
26	8, 13, 25	3imtr3i 294	. . 3 ⊢ (∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
27	7, 26	chvarvv 2021	. 2 ⊢ (∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
28	27	19.35ri 1911	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-rep 5231

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816

This theorem is used by:  axrep2  5234