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Theorem ax6e2eq 37794
 Description: Alternate form of ax6e 2238 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 37794 is derived from ax6e2eqVD 38165. (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 1877 . . . . . . 7 𝑥 𝑥 = 𝑢
2 hbae 2303 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
3 ax7 1930 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
43sps 2043 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢𝑦 = 𝑢))
54ancld 574 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
62, 5eximdh 1778 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢)))
71, 6mpi 20 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢))
87axc4i 2116 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
9 axc11 2302 . . . . 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 1879 . . . 4 (∀𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
1210, 11syl 17 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢))
13 excomim 2030 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
1412, 13syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢))
15 equtrr 1936 . . . 4 (𝑢 = 𝑣 → (𝑦 = 𝑢𝑦 = 𝑣))
1615anim2d 587 . . 3 (𝑢 = 𝑣 → ((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣)))
17162eximdv 1835 . 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 383  ∀wal 1473  ∃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 This theorem is referenced by:  ax6e2ndeq  37796  ax6e2ndeqVD  38167  ax6e2ndeqALT  38189
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