Theorem ax16 1691

Description: Theorem showing that ax-16 1692 is redundant if ax-17 1416 is included in the axiom system. The important part of the proof is provided by aev 1690.

See ax16ALT 1736 for an alternate proof that does not require ax-10 1393 or ax-12 1399.

This theorem should not be referenced in any proof. Instead, use ax-16 1692 below so that theorems needing ax-16 1692 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Assertion

Ref	Expression
ax16	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1690	. 2 ⊢ (∀x x = y → ∀z x = z)
2		ax-17 1416	. . . 4 ⊢ (φ → ∀zφ)
3		sbequ12 1651	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
4	3	biimpcd 148	. . . 4 ⊢ (φ → (x = z → [z / x]φ))
5	2, 4	alimdh 1353	. . 3 ⊢ (φ → (∀z x = z → ∀z[z / x]φ))
6	2	hbsb3 1686	. . . 4 ⊢ ([z / x]φ → ∀x[z / x]φ)
7		stdpc7 1650	. . . 4 ⊢ (z = x → ([z / x]φ → φ))
8	6, 2, 7	cbv3h 1628	. . 3 ⊢ (∀z[z / x]φ → ∀xφ)
9	5, 8	syl6com 31	. 2 ⊢ (∀z x = z → (φ → ∀xφ))
10	1, 9	syl 14	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 4  ∀wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  dveeq2  1693  dveeq2or  1694  a16g  1741  exists2  1994