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Theorem ax6e2ndeqVD 38628
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 38267) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 38255 is ax6e2ndeqVD 38628 without virtual deductions and was automatically derived from ax6e2ndeqVD 38628. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝑢𝑣   ▶   𝑢𝑣   )
2:: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   ( 𝑥 = 𝑢𝑦 = 𝑣)   )
3:2: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 = 𝑢   )
4:1,3: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 𝑣   )
5:2: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑦 = 𝑣   )
6:4,5: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 𝑦   )
7:: (∀𝑥𝑥 = 𝑦𝑥 = 𝑦)
8:7: 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
9:: 𝑥 = 𝑦𝑥𝑦)
10:8,9: (𝑥𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
11:6,10: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶    ¬ ∀𝑥𝑥 = 𝑦   )
12:11: (   𝑢𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
13:12: (   𝑢𝑣   ▶   𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
14:13: (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦)   )
15:: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 )
19:15: (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 = 𝑦)
20:14,19: (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
21:20: (   𝑢𝑣   ▶   𝑦(∃𝑥(𝑥 = 𝑢 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
22:21: (   𝑢𝑣   ▶   (∃𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
23:: (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃ 𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
24:22,23: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
25:: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 )
26:25: (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦𝑦¬ 𝑥𝑥 = 𝑦)
260:: (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦𝑦¬ 𝑥𝑥 = 𝑦)
27:260: (∃𝑦𝑦¬ ∀𝑥𝑥 = 𝑦 ↔ ∀𝑦¬ 𝑥𝑥 = 𝑦)
270:26,27: (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥 𝑥 = 𝑦)
28:: (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 )
29:270,28: (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 )
30:24,29: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
31:30: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))   )
32:31: (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣)))
33:: (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
34:33: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣)   )
35:34: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))   )
36:35: (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣)))
37:: (𝑢 = 𝑣𝑢𝑣)
38:32,36,37: (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ( ¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))
39:: (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦 (𝑥 = 𝑢𝑦 = 𝑣)))
40:: (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣))
41:40: (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
42:: (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦)
43:39,41,42: (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣 ))
44:40,43: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣))
qed:38,44: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Assertion
Ref Expression
ax6e2ndeqVD ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2ndeqVD
StepHypRef Expression
1 ax6e2nd 38256 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 38255 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31a1d 25 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4 exmid 431 . . . 4 (∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦)
5 jao 534 . . . 4 ((∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → ((¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → ((∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))))
62, 3, 4, 5e000 38476 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
71, 6jaoi 394 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
8 idn1 38272 . . . . . . . . . . . . . . . 16 (   𝑢𝑣   ▶   𝑢𝑣   )
9 idn2 38320 . . . . . . . . . . . . . . . . 17 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
10 simpl 473 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
119, 10e2 38338 . . . . . . . . . . . . . . . 16 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 = 𝑢   )
12 neeq1 2852 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → (𝑥𝑣𝑢𝑣))
1312biimprcd 240 . . . . . . . . . . . . . . . 16 (𝑢𝑣 → (𝑥 = 𝑢𝑥𝑣))
148, 11, 13e12 38433 . . . . . . . . . . . . . . 15 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥𝑣   )
15 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
169, 15e2 38338 . . . . . . . . . . . . . . 15 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑦 = 𝑣   )
17 neeq2 2853 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
1817biimprcd 240 . . . . . . . . . . . . . . 15 (𝑥𝑣 → (𝑦 = 𝑣𝑥𝑦))
1914, 16, 18e22 38378 . . . . . . . . . . . . . 14 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥𝑦   )
20 df-ne 2791 . . . . . . . . . . . . . . . 16 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
2120bicomi 214 . . . . . . . . . . . . . . 15 𝑥 = 𝑦𝑥𝑦)
22 sp 2051 . . . . . . . . . . . . . . . 16 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2322con3i 150 . . . . . . . . . . . . . . 15 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
2421, 23sylbir 225 . . . . . . . . . . . . . 14 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
2519, 24e2 38338 . . . . . . . . . . . . 13 (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶    ¬ ∀𝑥 𝑥 = 𝑦   )
2625in2 38312 . . . . . . . . . . . 12 (   𝑢𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)   )
2726gen11 38323 . . . . . . . . . . 11 (   𝑢𝑣   ▶   𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)   )
28 exim 1758 . . . . . . . . . . 11 (∀𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
2927, 28e1a 38334 . . . . . . . . . 10 (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦)   )
30 nfnae 2317 . . . . . . . . . . 11 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
313019.9 2070 . . . . . . . . . 10 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
32 imbi2 338 . . . . . . . . . . 11 ((∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦) ↔ (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)))
3332biimpcd 239 . . . . . . . . . 10 ((∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)))
3429, 31, 33e10 38401 . . . . . . . . 9 (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)   )
3534gen11 38323 . . . . . . . 8 (   𝑢𝑣   ▶   𝑦(∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)   )
36 exim 1758 . . . . . . . 8 (∀𝑦(∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
3735, 36e1a 38334 . . . . . . 7 (   𝑢𝑣   ▶   (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)   )
38 excom 2039 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
39 imbi1 337 . . . . . . . 8 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)) → ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)))
4039biimprcd 240 . . . . . . 7 ((∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)))
4137, 38, 40e10 38401 . . . . . 6 (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)   )
42 hbnae 2316 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
4342eximi 1759 . . . . . . . 8 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
44 nfa1 2025 . . . . . . . . 9 𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦
454419.9 2070 . . . . . . . 8 (∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
4643, 45sylib 208 . . . . . . 7 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
47 sp 2051 . . . . . . 7 (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
4846, 47syl 17 . . . . . 6 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
49 imim1 83 . . . . . 6 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)))
5041, 48, 49e10 38401 . . . . 5 (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)   )
51 orc 400 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
5251imim2i 16 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
5350, 52e1a 38334 . . . 4 (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))   )
5453in1 38269 . . 3 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
55 idn1 38272 . . . . . 6 (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
56 ax-1 6 . . . . . 6 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣))
5755, 56e1a 38334 . . . . 5 (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣)   )
58 olc 399 . . . . . 6 (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
5958imim2i 16 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
6057, 59e1a 38334 . . . 4 (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))   )
6160in1 38269 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
62 exmidne 2800 . . 3 (𝑢 = 𝑣𝑢𝑣)
63 jao 534 . . . 4 ((𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢 = 𝑣𝑢𝑣) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))))
6463com12 32 . . 3 ((𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢 = 𝑣𝑢𝑣) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))))
6554, 61, 62, 64e000 38476 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
667, 65impbii 199 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wne 2790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-v 3188  df-vd1 38268  df-vd2 38276
This theorem is referenced by: (None)
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