Theorem ax16ALT 1736

Description: Version of ax16 1691 that doesn't require ax-10 1393 or ax-12 1399 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax16ALT	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16ALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbequ12 1651	. 2 ⊢ (x = z → (φ ↔ [z / x]φ))
2		ax-17 1416	. . 3 ⊢ (φ → ∀zφ)
3	2	hbsb3 1686	. 2 ⊢ ([z / x]φ → ∀x[z / x]φ)
4	1, 3	ax16i 1735	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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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:  dvelimALT  1883  dvelimfv  1884