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Theorem ax6e2eqVD 38626
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 ax6e2eqVD 38626 without virtual deductions and was automatically derived from ax6e2eqVD 38626. (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 = 𝑢𝑦 = 𝑢)   )
6:5: (   𝑥𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢 𝑦 = 𝑢))   )
7:6: (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
8:7: (∀𝑥𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → ( 𝑥 = 𝑢𝑦 = 𝑢)))
9:: (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥𝑥𝑥 = 𝑦)
10:8,9: (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 𝑦 = 𝑢)))
11:1,10: (   𝑥𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
12:11: (   𝑥𝑥 = 𝑦   ▶   (∃𝑥𝑥 = 𝑢 → ∃𝑥 (𝑥 = 𝑢𝑦 = 𝑢))   )
13:: 𝑥𝑥 = 𝑢
14:13,12: (   𝑥𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢𝑦 = 𝑢 )   )
140:14: (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢) )
141:140: (∀𝑥𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
15:1,141: (   𝑥𝑥 = 𝑦   ▶   𝑥𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
16:1,15: (   𝑥𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
17:16: (   𝑥𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
18:17: (   𝑥𝑥 = 𝑦   ▶   𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑢)   )
19:: (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
20:: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
21:20: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑢    )
22:19,21: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑣    )
23:20: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑥 = 𝑢    )
24:22,23: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
25:24: (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑢) → ( 𝑥 = 𝑢𝑦 = 𝑣))   )
26:25: (   𝑢 = 𝑣   ▶   𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
27:26: (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
28:27: (   𝑢 = 𝑣   ▶   𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
29:28: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
30:29: (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢 ) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31:18,30: (   𝑥𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥𝑦 (𝑥 = 𝑢𝑦 = 𝑣))   )
qed:31: (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦( 𝑥 = 𝑢𝑦 = 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 38272 . . . . . 6 (   𝑥 𝑥 = 𝑦   ▶   𝑥 𝑥 = 𝑦   )
2 ax6ev 1887 . . . . . . . . . 10 𝑥 𝑥 = 𝑢
3 hba1 2148 . . . . . . . . . . . . . 14 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
4 sp 2051 . . . . . . . . . . . . . 14 (∀𝑥𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
53, 4impbii 199 . . . . . . . . . . . . 13 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑥 𝑥 = 𝑦)
6 idn2 38320 . . . . . . . . . . . . . . . . 17 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
7 sp 2051 . . . . . . . . . . . . . . . . . . 19 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
81, 7e1a 38334 . . . . . . . . . . . . . . . . . 18 (   𝑥 𝑥 = 𝑦   ▶   𝑥 = 𝑦   )
9 ax7 1940 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
109com12 32 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑢 → (𝑥 = 𝑦𝑦 = 𝑢))
116, 8, 10e21 38439 . . . . . . . . . . . . . . . . 17 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑦 = 𝑢   )
12 pm3.2 463 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → (𝑦 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
136, 11, 12e22 38378 . . . . . . . . . . . . . . . 16 (   𝑥 𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
1413in2 38312 . . . . . . . . . . . . . . 15 (   𝑥 𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
1514in1 38269 . . . . . . . . . . . . . 14 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
1615alimi 1736 . . . . . . . . . . . . 13 (∀𝑥𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
175, 16sylbi 207 . . . . . . . . . . . 12 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
181, 17e1a 38334 . . . . . . . . . . 11 (   𝑥 𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
19 exim 1758 . . . . . . . . . . 11 (∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)) → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
2018, 19e1a 38334 . . . . . . . . . 10 (   𝑥 𝑥 = 𝑦   ▶   (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))   )
21 pm2.27 42 . . . . . . . . . 10 (∃𝑥 𝑥 = 𝑢 → ((∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
222, 20, 21e01 38398 . . . . . . . . 9 (   𝑥 𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
2322in1 38269 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))
2423axc4i 2127 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
251, 24e1a 38334 . . . . . 6 (   𝑥 𝑥 = 𝑦   ▶   𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
26 axc11 2313 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
271, 25, 26e11 38395 . . . . 5 (   𝑥 𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
28 19.2 1889 . . . . 5 (∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
2927, 28e1a 38334 . . . 4 (   𝑥 𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)   )
30 excomim 2040 . . . 4 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
3129, 30e1a 38334 . . 3 (   𝑥 𝑥 = 𝑦   ▶   𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢)   )
32 idn1 38272 . . . . . . . . . . 11 (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
33 idn2 38320 . . . . . . . . . . . 12 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
34 simpr 477 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑦 = 𝑢)
3533, 34e2 38338 . . . . . . . . . . 11 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑢   )
36 equtrr 1946 . . . . . . . . . . 11 (𝑢 = 𝑣 → (𝑦 = 𝑢𝑦 = 𝑣))
3732, 35, 36e12 38433 . . . . . . . . . 10 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑣   )
38 simpl 473 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑥 = 𝑢)
3933, 38e2 38338 . . . . . . . . . 10 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑥 = 𝑢   )
40 pm3.21 464 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑣)))
4137, 39, 40e22 38378 . . . . . . . . 9 (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4241in2 38312 . . . . . . . 8 (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
4342gen11 38323 . . . . . . 7 (   𝑢 = 𝑣   ▶   𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
44 exim 1758 . . . . . . 7 (∀𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4543, 44e1a 38334 . . . . . 6 (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
4645gen11 38323 . . . . 5 (   𝑢 = 𝑣   ▶   𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
47 exim 1758 . . . . 5 (∀𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4846, 47e1a 38334 . . . 4 (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
4948in1 38269 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
50 pm2.04 90 . . . 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 38401 . 2 (   𝑥 𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
5352in1 38269 1 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wex 1701
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-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-vd1 38268  df-vd2 38276
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator