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Theorem a16g 1741
Description: A generalization of axiom ax-16 1692. (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 1690 . 2 (x x = yz z = x)
2 ax16 1691 . 2 (x x = y → (φxφ))
3 biidd 161 . . . 4 (z z = x → (φφ))
43dral1 1615 . . 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 1240
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  a16gb  1742  a16nf  1743
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