Theorem axrepnd 10281

Description: A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

Assertion

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

Proof of Theorem axrepnd

Step	Hyp	Ref	Expression
1		axrepndlem2 10280	. . . 4 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
2		nfnae 2434	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3		nfnae 2434	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑧
4	2, 3	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5		nfnae 2434	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
6	4, 5	nfan 1903	. . . . 5 ⊢ Ⅎ𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
7		nfnae 2434	. . . . . . . . 9 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
8		nfnae 2434	. . . . . . . . 9 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑧
9	7, 8	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
10		nfnae 2434	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
11	9, 10	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
12		nfcvf 2935	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
13	12	adantl 481	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧)
14		nfcvf2 2936	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
15	14	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥)
16	13, 15	nfeld 2917	. . . . . . . . . 10 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧 ∈ 𝑥)
17	16	nf5rd 2192	. . . . . . . . 9 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧 ∈ 𝑥 → ∀𝑦 𝑧 ∈ 𝑥))
18		sp 2178	. . . . . . . . 9 ⊢ (∀𝑦 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑥)
19	17, 18	impbid1 224	. . . . . . . 8 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧 ∈ 𝑥 ↔ ∀𝑦 𝑧 ∈ 𝑥))
20		nfcvf2 2936	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑧𝑥)
21	20	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑥)
22		nfcvf2 2936	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
23	22	adantl 481	. . . . . . . . . . . . 13 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑦)
24	21, 23	nfeld 2917	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑥 ∈ 𝑦)
25	24	nf5rd 2192	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
26		sp 2178	. . . . . . . . . . 11 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦)
27	25, 26	impbid1 224	. . . . . . . . . 10 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥 ∈ 𝑦 ↔ ∀𝑧 𝑥 ∈ 𝑦))
28	27	anbi1d 629	. . . . . . . . 9 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑) ↔ (∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
29	6, 28	exbid 2219	. . . . . . . 8 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
30	19, 29	bibi12d 345	. . . . . . 7 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)) ↔ (∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
31	11, 30	albid 2218	. . . . . 6 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
32	31	imbi2d 340	. . . . 5 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))))
33	6, 32	exbid 2219	. . . 4 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))))
34	1, 33	mpbid 231	. . 3 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
35	34	exp31 419	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))))
36		nfae 2433	. . . . 5 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
37		nd2 10275	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑧 ∈ 𝑥)
38	37	aecoms 2428	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑧 ∈ 𝑥)
39		nfae 2433	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
40		nd3 10276	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
41	40	intnanrd 489	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ (∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
42	39, 41	nexd 2217	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
43	38, 42	2falsed 376	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
44	36, 43	alrimi 2209	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
45	44	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
46	45	19.8ad 2177	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
47		nfae 2433	. . . . 5 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑧
48		nd4 10277	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑧 ∈ 𝑥)
49		nfae 2433	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧
50		nd1 10274	. . . . . . . . 9 ⊢ (∀𝑧 𝑧 = 𝑥 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
51	50	aecoms 2428	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
52	51	intnanrd 489	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ (∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
53	49, 52	nexd 2217	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
54	48, 53	2falsed 376	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
55	47, 54	alrimi 2209	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
56	55	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
57	56	19.8ad 2177	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
58		nfae 2433	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 𝑦 = 𝑧
59		nd1 10274	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑧 ∈ 𝑥)
60		nfae 2433	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑧
61		nd2 10275	. . . . . . . . 9 ⊢ (∀𝑧 𝑧 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
62	61	aecoms 2428	. . . . . . . 8 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
63	62	intnanrd 489	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ (∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
64	60, 63	nexd 2217	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))
65	59, 64	2falsed 376	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
66	58, 65	alrimi 2209	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
67	66	a1d 25	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
68	67	19.8ad 2177	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
69	35, 46, 57, 68	pm2.61iii 185	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  zfcndrep  10301  axrepprim  33543