Theorem ax16ALT 2047

Description: Alternate proof of ax16 2045. (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 1919	. 2 ⊢ (x = z → (φ ↔ [z / x]φ))
2		ax-17 1616	. . 3 ⊢ (φ → ∀zφ)
3	2	hbsb3 2043	. 2 ⊢ ([z / x]φ → ∀x[z / x]φ)
4	1, 3	ax16i 2046	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648

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  df-sb 1649

This theorem is referenced by:  dvelimALT  2133