Theorem ax6e2eq 45307

Description: Alternate form of ax6e 2418 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 45307 is derived from ax6e2eqVD 45656. (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 2002	. . . . . . 7 ⊢ ∃𝑥 𝑥 = 𝑢
2		hbae 2466	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑥 𝑥 = 𝑦)
3		ax7 2049	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑢 → 𝑦 = 𝑢))
4	3	sps 2224	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → 𝑦 = 𝑢))
5	4	ancld 560	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
6	2, 5	eximdh 1897	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑢 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
7	1, 6	mpi 21	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
8	7	axc4i 2358	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
9		axc11 2465	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
10	8, 9	mpd 16	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
11		19.2 2009	. . . 4 ⊢ (∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
12	10, 11	syl 18	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
13		excomim 2201	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
14	12, 13	syl 18	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
15		equtrr 2055	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑦 = 𝑢 → 𝑦 = 𝑣))
16	15	anim2d 624	. . 3 ⊢ (𝑢 = 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
17	16	2eximdv 1952	. 2 ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
18	14, 17	syl5com 32	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  ax6e2ndeq  45309  ax6e2ndeqVD  45658  ax6e2ndeqALT  45680