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Theorem dvelimALT 2133
 Description: Version of dvelim 2016 that doesn't use ax-10 2140. (See dvelimh 1964 for a version that doesn't use ax-11 1746.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dvelimALT.1 (φxφ)
dvelimALT.2 (z = y → (φψ))
Assertion
Ref Expression
dvelimALT x x = y → (ψxψ))
Distinct variable groups:   ψ,z   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelimALT
StepHypRef Expression
1 ax-17 1616 . . 3 x x = yz ¬ x x = y)
2 ax16ALT 2047 . . . . 5 (x x = z → ((z = yφ) → x(z = yφ)))
32a1d 22 . . . 4 (x x = z → (¬ x x = y → ((z = yφ) → x(z = yφ))))
4 hbn1 1730 . . . . . . 7 x x = zx ¬ x x = z)
5 hbn1 1730 . . . . . . 7 x x = yx ¬ x x = y)
64, 5hban 1828 . . . . . 6 ((¬ x x = z ¬ x x = y) → xx x = z ¬ x x = y))
7 ax12o 1934 . . . . . . 7 x x = z → (¬ x x = y → (z = yx z = y)))
87imp 418 . . . . . 6 ((¬ x x = z ¬ x x = y) → (z = yx z = y))
9 dvelimALT.1 . . . . . . 7 (φxφ)
109a1i 10 . . . . . 6 ((¬ x x = z ¬ x x = y) → (φxφ))
116, 8, 10hbimd 1815 . . . . 5 ((¬ x x = z ¬ x x = y) → ((z = yφ) → x(z = yφ)))
1211ex 423 . . . 4 x x = z → (¬ x x = y → ((z = yφ) → x(z = yφ))))
133, 12pm2.61i 156 . . 3 x x = y → ((z = yφ) → x(z = yφ)))
141, 13hbald 1740 . 2 x x = y → (z(z = yφ) → xz(z = yφ)))
15 ax-17 1616 . . 3 (ψzψ)
16 dvelimALT.2 . . 3 (z = y → (φψ))
1715, 16equsalh 1961 . 2 (z(z = yφ) ↔ ψ)
1817albii 1566 . 2 (xz(z = yφ) ↔ xψ)
1914, 17, 183imtr3g 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-1 5  ax-2 6  ax-3 7  ax-mp 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  df-sb 1649 This theorem is referenced by:  dveeq2-o16  2185
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