Theorem ax16i 2046

Description: Inference with ax16 2045 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

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		nfv 1619	. . 3 ⊢ Ⅎz x = y
2		nfv 1619	. . 3 ⊢ Ⅎx z = y
3		ax-8 1675	. . 3 ⊢ (x = z → (x = y → z = y))
4	1, 2, 3	cbv3 1982	. 2 ⊢ (∀x x = y → ∀z z = y)
5		ax-8 1675	. . . . 5 ⊢ (z = x → (z = y → x = y))
6	5	spimv 1990	. . . 4 ⊢ (∀z z = y → x = y)
7		equcomi 1679	. . . . . 6 ⊢ (x = y → y = x)
8		equcomi 1679	. . . . . . 7 ⊢ (z = y → y = z)
9		ax-8 1675	. . . . . . 7 ⊢ (y = z → (y = x → z = x))
10	8, 9	syl 15	. . . . . 6 ⊢ (z = y → (y = x → z = x))
11	7, 10	syl5com 26	. . . . 5 ⊢ (x = y → (z = y → z = x))
12	11	alimdv 1621	. . . 4 ⊢ (x = y → (∀z z = y → ∀z z = x))
13	6, 12	mpcom 32	. . 3 ⊢ (∀z z = y → ∀z z = x)
14		equcomi 1679	. . . 4 ⊢ (z = x → x = z)
15	14	alimi 1559	. . 3 ⊢ (∀z z = x → ∀z x = z)
16	13, 15	syl 15	. 2 ⊢ (∀z z = y → ∀z x = z)
17		ax16i.1	. . . . 5 ⊢ (x = z → (φ ↔ ψ))
18	17	biimpcd 215	. . . 4 ⊢ (φ → (x = z → ψ))
19	18	alimdv 1621	. . 3 ⊢ (φ → (∀z x = z → ∀zψ))
20		ax16i.2	. . . . 5 ⊢ (ψ → ∀xψ)
21	20	nfi 1551	. . . 4 ⊢ Ⅎxψ
22		nfv 1619	. . . 4 ⊢ Ⅎzφ
23	17	biimprd 214	. . . . 5 ⊢ (x = z → (ψ → φ))
24	14, 23	syl 15	. . . 4 ⊢ (z = x → (ψ → φ))
25	21, 22, 24	cbv3 1982	. . 3 ⊢ (∀zψ → ∀xφ)
26	19, 25	syl6com 31	. 2 ⊢ (∀z x = z → (φ → ∀xφ))
27	4, 16, 26	3syl 18	1 ⊢ (∀x x = y → (φ → ∀xφ))

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

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

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:  ax16ALT  2047