Theorem axrep1 5155

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 5154 → axrep1 5155 → axrep2 5157 → axrepnd 10005 → zfcndrep 10025 = ax-rep 5154. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2379. (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 2126	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
2	1	anbi1d 632	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)))
3	2	exbidv 1922	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
4	3	bibi2d 346	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
5	4	albidv 1921	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
6	5	exbidv 1922	. . . 4 ⊢ (𝑤 = 𝑦 → (∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))))
7	6	imbi2d 344	. . 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 5154	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
9		19.3v 1986	. . . . . . . 8 ⊢ (∀𝑦𝜑 ↔ 𝜑)
10	9	imbi1i 353	. . . . . . 7 ⊢ ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (𝜑 → 𝑧 = 𝑦))
11	10	albii 1821	. . . . . 6 ⊢ (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(𝜑 → 𝑧 = 𝑦))
12	11	exbii 1849	. . . . 5 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
13	12	albii 1821	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
14		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑦
15		nfe1 2151	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
16	14, 15	nfbi 1904	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
17	16	nfal 2331	. . . . 5 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
18		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
19		elequ2 2126	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥))
20	9	anbi2i 625	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
21	20	exbii 1849	. . . . . . . 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 349	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
24	23	albidv 1921	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
25	17, 18, 24	cbvexv1 2351	. . . 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 2005	. 2 ⊢ (∀𝑥∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
28	27	19.35ri 1880	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-rep 5154

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by:  axrep2  5157