Theorem ax10-16 2190

Description: This theorem shows that, given ax-16 2144, we can derive a version of ax-10 2140. However, it is weaker than ax-10 2140 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10-16	⊢ (∀x x = z → ∀z z = x)

Distinct variable group:   x,z

Proof of Theorem ax10-16

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-16 2144	. . . 4 ⊢ (∀x x = z → (x = w → ∀x x = w))
2	1	alrimiv 1631	. . 3 ⊢ (∀x x = z → ∀w(x = w → ∀x x = w))
3	2	a5i-o 2150	. 2 ⊢ (∀x x = z → ∀x∀w(x = w → ∀x x = w))
4		equequ1 1684	. . . . . 6 ⊢ (x = z → (x = w ↔ z = w))
5	4	cbvalv 2002	. . . . . . 7 ⊢ (∀x x = w ↔ ∀z z = w)
6	5	a1i 10	. . . . . 6 ⊢ (x = z → (∀x x = w ↔ ∀z z = w))
7	4, 6	imbi12d 311	. . . . 5 ⊢ (x = z → ((x = w → ∀x x = w) ↔ (z = w → ∀z z = w)))
8	7	albidv 1625	. . . 4 ⊢ (x = z → (∀w(x = w → ∀x x = w) ↔ ∀w(z = w → ∀z z = w)))
9	8	cbvalv 2002	. . 3 ⊢ (∀x∀w(x = w → ∀x x = w) ↔ ∀z∀w(z = w → ∀z z = w))
10	9	biimpi 186	. 2 ⊢ (∀x∀w(x = w → ∀x x = w) → ∀z∀w(z = w → ∀z z = w))
11		nfa1-o 2166	. . . . . . 7 ⊢ Ⅎz∀z z = w
12	11	19.23 1801	. . . . . 6 ⊢ (∀z(z = w → ∀z z = w) ↔ (∃z z = w → ∀z z = w))
13	12	albii 1566	. . . . 5 ⊢ (∀w∀z(z = w → ∀z z = w) ↔ ∀w(∃z z = w → ∀z z = w))
14		a9ev 1656	. . . . . . . 8 ⊢ ∃z z = w
15		pm2.27 35	. . . . . . . 8 ⊢ (∃z z = w → ((∃z z = w → ∀z z = w) → ∀z z = w))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((∃z z = w → ∀z z = w) → ∀z z = w)
17	16	alimi 1559	. . . . . 6 ⊢ (∀w(∃z z = w → ∀z z = w) → ∀w∀z z = w)
18		equequ2 1686	. . . . . . . . 9 ⊢ (w = x → (z = w ↔ z = x))
19	18	spv 1998	. . . . . . . 8 ⊢ (∀w z = w → z = x)
20	19	sps-o 2159	. . . . . . 7 ⊢ (∀z∀w z = w → z = x)
21	20	a7s 1735	. . . . . 6 ⊢ (∀w∀z z = w → z = x)
22	17, 21	syl 15	. . . . 5 ⊢ (∀w(∃z z = w → ∀z z = w) → z = x)
23	13, 22	sylbi 187	. . . 4 ⊢ (∀w∀z(z = w → ∀z z = w) → z = x)
24	23	a7s 1735	. . 3 ⊢ (∀z∀w(z = w → ∀z z = w) → z = x)
25	24	a5i-o 2150	. 2 ⊢ (∀z∀w(z = w → ∀z z = w) → ∀z z = x)
26	3, 10, 25	3syl 18	1 ⊢ (∀x x = z → ∀z z = x)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-16 2144

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)