Theorem ax6e2eq 42177

Description: Alternate form of ax6e 2383 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 42177 is derived from ax6e2eqVD 42527. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2eq	⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2eq

Step	Hyp	Ref	Expression
1		ax6ev 1973	. . . . . . 7 ⊢ ∃𝑥 𝑥 = 𝑢
2		hbae 2431	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑥 𝑥 = 𝑦)
3		ax7 2019	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑢 → 𝑦 = 𝑢))
4	3	sps 2178	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → 𝑦 = 𝑢))
5	4	ancld 551	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
6	2, 5	eximdh 1867	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
7	1, 6	mpi 20	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
8	7	axc4i 2316	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
9		axc11 2430	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
10	8, 9	mpd 15	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
11		19.2 1980	. . . 4 ⊢ (∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
12	10, 11	syl 17	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
13		excomim 2163	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
14	12, 13	syl 17	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
15		equtrr 2025	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑦 = 𝑢 → 𝑦 = 𝑣))
16	15	anim2d 612	. . 3 ⊢ (𝑢 = 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
17	16	2eximdv 1922	. 2 ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
18	14, 17	syl5com 31	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  ax6e2ndeq  42179  ax6e2ndeqVD  42529  ax6e2ndeqALT  42551