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Theorem ax6e2eq 39435
Description: Alternate form of ax6e 2356 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 39435 is derived from ax6e2eqVD 39795. (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 2072 . . . . . . 7 𝑥 𝑥 = 𝑢
2 hbae 2411 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
3 ax7 2113 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
43sps 2217 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
54ancld 546 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
62, 5eximdh 1961 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
71, 6mpi 20 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))
87axc4i 2308 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
9 axc11 2410 . . . . 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 2074 . . . 4 (∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
1210, 11syl 17 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
13 excomim 2207 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
1412, 13syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
15 equtrr 2119 . . . 4 (𝑢 = 𝑣 → (𝑦 = 𝑢𝑦 = 𝑣))
1615anim2d 605 . . 3 (𝑢 = 𝑣 → ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣)))
17162eximdv 2014 . 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 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879
This theorem is referenced by:  ax6e2ndeq  39437  ax6e2ndeqVD  39797  ax6e2ndeqALT  39819
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