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Theorem a16g-o 2186
 Description: A generalization of axiom ax-16 2144. Version of a16g 1945 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a16g-o (x x = y → (φzφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem a16g-o
StepHypRef Expression
1 aev-o 2182 . 2 (x x = yz z = x)
2 ax-16 2144 . 2 (x x = y → (φxφ))
3 biidd 228 . . . 4 (z z = x → (φφ))
43dral1-o 2154 . . 3 (z z = x → (zφxφ))
54biimprd 214 . 2 (z z = x → (xφzφ))
61, 2, 5sylsyld 52 1 (x x = y → (φzφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax11inda2  2199
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