Theorem ax16i 1851

Description: Inference with ax-16 1807 as its conclusion, that does not require ax-10 1498, ax-11 1499, or ax12 1505 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)

Hypotheses

Ref	Expression
ax16i.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
ax16i.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
ax16i	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem ax16i

Step	Hyp	Ref	Expression
1		ax-17 1519	. . . 4 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
2		ax-17 1519	. . . 4 ⊢ (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
3		ax-8 1497	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
4	1, 2, 3	cbv3h 1736	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
5		ax-8 1497	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
6	5	spimv 1804	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → 𝑥 = 𝑦)
7		equid 1694	. . . . . . . 8 ⊢ 𝑥 = 𝑥
8		ax-8 1497	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥))
9	7, 8	mpi 15	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
10		equid 1694	. . . . . . . . 9 ⊢ 𝑧 = 𝑧
11		ax-8 1497	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑧 → 𝑦 = 𝑧))
12	10, 11	mpi 15	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → 𝑦 = 𝑧)
13		ax-8 1497	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑥 → 𝑧 = 𝑥))
14	12, 13	syl 14	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑦 = 𝑥 → 𝑧 = 𝑥))
15	9, 14	syl5com 29	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
16	1, 15	alimdh 1460	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
17	6, 16	mpcom 36	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
18		ax-8 1497	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑧 → 𝑥 = 𝑧))
19	10, 18	mpi 15	. . . . 5 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧)
20	19	alimi 1448	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
21	17, 20	syl 14	. . 3 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
22	4, 21	syl 14	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
23		ax-17 1519	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
24		ax16i.1	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
25	24	biimpcd 158	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑧 → 𝜓))
26	23, 25	alimdh 1460	. . 3 ⊢ (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧𝜓))
27		ax16i.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
28	24	biimprd 157	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 → 𝜑))
29	19, 28	syl 14	. . . 4 ⊢ (𝑧 = 𝑥 → (𝜓 → 𝜑))
30	27, 23, 29	cbv3h 1736	. . 3 ⊢ (∀𝑧𝜓 → ∀𝑥𝜑)
31	26, 30	syl6com 35	. 2 ⊢ (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
32	22, 31	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  ax16ALT  1852