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Theorem a16g 1945
 Description: Generalization of ax16 2045. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
a16g (x x = y → (φzφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem a16g
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 a9ev 1656 . 2 w w = z
2 ax10lem5 1942 . 2 (x x = yw w = z)
3 hbn1 1730 . . . . 5 w w = zw ¬ w w = z)
4 pm2.21 100 . . . . 5 w w = z → (w w = z → (φzφ)))
53, 4alrimih 1565 . . . 4 w w = zw(w w = z → (φzφ)))
6 ax-17 1616 . . . . 5 ((φzφ) → w(φzφ))
7 ax-1 6 . . . . 5 ((φzφ) → (w w = z → (φzφ)))
86, 7alrimih 1565 . . . 4 ((φzφ) → w(w w = z → (φzφ)))
95, 8ja 153 . . 3 ((w w = z → (φzφ)) → w(w w = z → (φzφ)))
10 ax10lem5 1942 . . . 4 (w w = zz z = w)
11 equcomi 1679 . . . . . . 7 (w = zz = w)
12 ax-17 1616 . . . . . . 7 (φwφ)
13 ax-11 1746 . . . . . . 7 (z = w → (wφz(z = wφ)))
1411, 12, 13syl2im 34 . . . . . 6 (w = z → (φz(z = wφ)))
15 ax-5 1557 . . . . . 6 (z(z = wφ) → (z z = wzφ))
1614, 15syl6 29 . . . . 5 (w = z → (φ → (z z = wzφ)))
1716com23 72 . . . 4 (w = z → (z z = w → (φzφ)))
1810, 17syl5 28 . . 3 (w = z → (w w = z → (φzφ)))
199, 18exlimih 1804 . 2 (w w = z → (w w = z → (φzφ)))
201, 2, 19mpsyl 59 1 (x x = y → (φzφ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541 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 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:  ax16  2045  a16gb  2050  a16nf  2051
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