Theorem ax16ALT 1717

Description: Version of ax16 1672 that doesn't require ax-10 1373 or ax-12 1379 for its proof. (Contributed by NM, 17-May-2008.)

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 1632	. 2 ⊢ (x = z → (φ ↔ [z / x]φ))
2		ax-17 1396	. . 3 ⊢ (φ → ∀zφ)
3	2	hbsb3 1667	. 2 ⊢ ([z / x]φ → ∀x[z / x]φ)
4	1, 3	ax16i 1716	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 4  ∀wal 1224  [wsb 1623

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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624

This theorem is referenced by:  dvelimALT  1864  dvelimfv  1865