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Theorem ax6e2eq 38290
Description: Alternate form of ax6e 2249 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 38290 is derived from ax6e2eqVD 38661. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2eq (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2eq
StepHypRef Expression
1 ax6ev 1887 . . . . . . 7 𝑥 𝑥 = 𝑢
2 hbae 2314 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
3 ax7 1940 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
43sps 2053 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
54ancld 575 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
62, 5eximdh 1788 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
71, 6mpi 20 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))
87axc4i 2127 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
9 axc11 2313 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
108, 9mpd 15 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
11 19.2 1889 . . . 4 (∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
1210, 11syl 17 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
13 excomim 2040 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
1412, 13syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
15 equtrr 1946 . . . 4 (𝑢 = 𝑣 → (𝑦 = 𝑢𝑦 = 𝑣))
1615anim2d 588 . . 3 (𝑢 = 𝑣 → ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣)))
17162eximdv 1845 . 2 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
1814, 17syl5com 31 1 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  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-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  ax6e2ndeq  38292  ax6e2ndeqVD  38663  ax6e2ndeqALT  38685
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