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Theorem a16g 1717
 Description: A generalization of axiom ax-16 1668. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
a16g (x x = y → (φzφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem a16g
StepHypRef Expression
1 aev 1666 . 2 (x x = yz z = x)
2 ax16 1667 . 2 (x x = y → (φxφ))
3 biidd 161 . . . 4 (z z = x → (φφ))
43dral1 1591 . . 3 (z z = x → (zφxφ))
54biimprd 147 . 2 (z z = x → (xφzφ))
61, 2, 5sylsyld 52 1 (x x = y → (φzφ))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1221 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-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400 This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619 This theorem is referenced by:  a16gb  1718  a16nf  1719
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