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	|- (A.x x = y -> (ph -> A.xph))

Distinct variable group:   x,y

Allowed substitution hints:   ph(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 -> (ph <-> [z / x]ph))
2		ax-17 1616	. . 3 |- (ph -> A.zph)
3	2	hbsb3 2043	. 2 |- ([z / x]ph -> A.x[z / x]ph)
4	1, 3	ax16i 2046	1 |- (A.x x = y -> (ph -> A.xph))

This proof depends on syntax axioms:   -> wi 4  A.wal 1540  [wsb 1648

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is used by:  dvelimALT  2133