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Theorem dvelimf-o 2180
 Description: Proof of dvelimh 1964 that uses ax-10o 2139 but not ax-11o 2141, ax-10 2140, or ax-11 1746. Version of dvelimh 1964 using ax-10o 2139 instead of ax10o 1952. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimf-o.1 (φxφ)
dvelimf-o.2 (ψzψ)
dvelimf-o.3 (z = y → (φψ))
Assertion
Ref Expression
dvelimf-o x x = y → (ψxψ))

Proof of Theorem dvelimf-o
StepHypRef Expression
1 hba1-o 2149 . . . . 5 (z(z = yφ) → zz(z = yφ))
2 ax-10o 2139 . . . . . 6 (z z = x → (zz(z = yφ) → xz(z = yφ)))
32aecoms-o 2152 . . . . 5 (x x = z → (zz(z = yφ) → xz(z = yφ)))
41, 3syl5 28 . . . 4 (x x = z → (z(z = yφ) → xz(z = yφ)))
54a1d 22 . . 3 (x x = z → (¬ x x = y → (z(z = yφ) → xz(z = yφ))))
6 hbnae-o 2179 . . . . . 6 x x = zz ¬ x x = z)
7 hbnae-o 2179 . . . . . 6 x x = yz ¬ x x = y)
86, 7hban 1828 . . . . 5 ((¬ x x = z ¬ x x = y) → zx x = z ¬ x x = y))
9 hbnae-o 2179 . . . . . . 7 x x = zx ¬ x x = z)
10 hbnae-o 2179 . . . . . . 7 x x = yx ¬ x x = y)
119, 10hban 1828 . . . . . 6 ((¬ x x = z ¬ x x = y) → xx x = z ¬ x x = y))
12 ax-12o 2142 . . . . . . 7 x x = z → (¬ x x = y → (z = yx z = y)))
1312imp 418 . . . . . 6 ((¬ x x = z ¬ x x = y) → (z = yx z = y))
14 dvelimf-o.1 . . . . . . 7 (φxφ)
1514a1i 10 . . . . . 6 ((¬ x x = z ¬ x x = y) → (φxφ))
1611, 13, 15hbimd 1815 . . . . 5 ((¬ x x = z ¬ x x = y) → ((z = yφ) → x(z = yφ)))
178, 16hbald 1740 . . . 4 ((¬ x x = z ¬ x x = y) → (z(z = yφ) → xz(z = yφ)))
1817ex 423 . . 3 x x = z → (¬ x x = y → (z(z = yφ) → xz(z = yφ))))
195, 18pm2.61i 156 . 2 x x = y → (z(z = yφ) → xz(z = yφ)))
20 dvelimf-o.2 . . 3 (ψzψ)
21 dvelimf-o.3 . . 3 (z = y → (φψ))
2220, 21equsalh 1961 . 2 (z(z = yφ) ↔ ψ)
2322albii 1566 . 2 (xz(z = yφ) ↔ xψ)
2419, 22, 233imtr3g 260 1 x x = y → (ψxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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 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:  dveeq2-o  2184  dveeq1-o  2187  ax11el  2194
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