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Theorem ax6e2eqVD 38165
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 37806) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 37794 is ax6e2eqVD 38165 without virtual deductions and was automatically derived from ax6e2eqVD 38165. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥𝑥 = 𝑦   ) 2:: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   ) 3:1: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   𝑥 = 𝑦   ) 4:2,3: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑦 = 𝑢   ) 5:2,4: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 6:5: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 7:6: ⊢ (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 8:7: ⊢ (∀𝑥∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 9:: ⊢ (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥∀𝑥𝑥 = 𝑦) 10:8,9: ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 11:1,10: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 12:11: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∃𝑥𝑥 = 𝑢 → ∃𝑥 (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 13:: ⊢ ∃𝑥𝑥 = 𝑢 14:13,12: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 )   ) 140:14: ⊢ (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ) 141:140: ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)) 15:1,141: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 16:1,15: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 17:16: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 18:17: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 19:: ⊢ (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   ) 20:: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 21:20: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑢    ) 22:19,21: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑣    ) 23:20: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑥 = 𝑢    ) 24:22,23: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 25:24: ⊢ (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 26:25: ⊢ (   𝑢 = 𝑣   ▶   ∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 27:26: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 28:27: ⊢ (   𝑢 = 𝑣   ▶   ∀𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 29:28: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 30:29: ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 ) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 31:18,30: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥∃𝑦 (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) qed:31: ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
Assertion
Ref Expression
ax6e2eqVD (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2eqVD
StepHypRef Expression
1 idn1 37811 . . . . . 6 (   𝑥 𝑥 = 𝑦   ▶   𝑥 𝑥 = 𝑦   )
2 ax6ev 1877 . . . . . . . . . 10 𝑥 𝑥 = 𝑢
3 hba1 2137 . . . . . . . . . . . . . 14 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
4 sp 2041 . . . . . . . . . . . . . 14 (∀𝑥𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
53, 4impbii 198 . . . . . . . . . . . . 13 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑥 𝑥 = 𝑦)
6 idn2 37859 . . . . . . . . . . . . . . . . 17 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
7 sp 2041 . . . . . . . . . . . . . . . . . . 19 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
81, 7e1a 37873 . . . . . . . . . . . . . . . . . 18 (   𝑥 𝑥 = 𝑦   ▶   𝑥 = 𝑦   )
9 ax7 1930 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
109com12 32 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑢 → (𝑥 = 𝑦𝑦 = 𝑢))
116, 8, 10e21 37978 . . . . . . . . . . . . . . . . 17 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑦 = 𝑢   )
12 pm3.2 462 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → (𝑦 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
136, 11, 12e22 37917 . . . . . . . . . . . . . . . 16 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
1413in2 37851 . . . . . . . . . . . . . . 15 (   𝑥 𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
1514in1 37808 . . . . . . . . . . . . . 14 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
1615alimi 1730 . . . . . . . . . . . . 13 (∀𝑥𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
175, 16sylbi 206 . . . . . . . . . . . 12 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
181, 17e1a 37873 . . . . . . . . . . 11 (   𝑥 𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
19 exim 1751 . . . . . . . . . . 11 (∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)) → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
2018, 19e1a 37873 . . . . . . . . . 10 (   𝑥 𝑥 = 𝑦   ▶   (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))   )
21 pm2.27 41 . . . . . . . . . 10 (∃𝑥 𝑥 = 𝑢 → ((∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
222, 20, 21e01 37937 . . . . . . . . 9 (   𝑥 𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
2322in1 37808 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))
2423axc4i 2116 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
251, 24e1a 37873 . . . . . 6 (   𝑥 𝑥 = 𝑦   ▶   𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
26 axc11 2302 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
271, 25, 26e11 37934 . . . . 5 (   𝑥 𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
28 19.2 1879 . . . . 5 (∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
2927, 28e1a 37873 . . . 4 (   𝑥 𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
30 excomim 2030 . . . 4 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
3129, 30e1a 37873 . . 3 (   𝑥 𝑥 = 𝑦   ▶   𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢)   )
32 idn1 37811 . . . . . . . . . . 11 (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
33 idn2 37859 . . . . . . . . . . . 12 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
34 simpr 476 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑦 = 𝑢)
3533, 34e2 37877 . . . . . . . . . . 11 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑢   )
36 equtrr 1936 . . . . . . . . . . 11 (𝑢 = 𝑣 → (𝑦 = 𝑢𝑦 = 𝑣))
3732, 35, 36e12 37972 . . . . . . . . . 10 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑣   )
38 simpl 472 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑥 = 𝑢)
3933, 38e2 37877 . . . . . . . . . 10 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑥 = 𝑢   )
40 pm3.21 463 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑣)))
4137, 39, 40e22 37917 . . . . . . . . 9 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4241in2 37851 . . . . . . . 8 (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
4342gen11 37862 . . . . . . 7 (   𝑢 = 𝑣   ▶   𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
44 exim 1751 . . . . . . 7 (∀𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4543, 44e1a 37873 . . . . . 6 (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
4645gen11 37862 . . . . 5 (   𝑢 = 𝑣   ▶   𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
47 exim 1751 . . . . 5 (∀𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4846, 47e1a 37873 . . . 4 (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
4948in1 37808 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
50 pm2.04 88 . . . 4 ((𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))))
5150com12 32 . . 3 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ((𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))))
5231, 49, 51e10 37940 . 2 (   𝑥 𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
5352in1 37808 1 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-vd1 37807  df-vd2 37815 This theorem is referenced by: (None)
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