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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 → (φxφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16ALT
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbequ12 1919 . 2 (x = z → (φ ↔ [z / x]φ))
2 ax-17 1616 . . 3 (φzφ)
32hbsb3 2043 . 2 ([z / x]φx[z / x]φ)
41, 3ax16i 2046 1 (x x = y → (φxφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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