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Theorem dvelimhw 1849
 Description: Proof of dvelimh 1964 without using ax-12 1925 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)
Hypotheses
Ref Expression
dvelimhw.1 (φxφ)
dvelimhw.2 (ψzψ)
dvelimhw.3 (z = y → (φψ))
dvelimhw.4 x x = y → (y = zx y = z))
Assertion
Ref Expression
dvelimhw x x = y → (ψxψ))
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem dvelimhw
StepHypRef Expression
1 ax-17 1616 . . 3 x x = yz ¬ x x = y)
2 hbn1 1730 . . . 4 x x = yx ¬ x x = y)
3 equcomi 1679 . . . . 5 (z = yy = z)
4 dvelimhw.4 . . . . 5 x x = y → (y = zx y = z))
5 equcomi 1679 . . . . . 6 (y = zz = y)
65alimi 1559 . . . . 5 (x y = zx z = y)
73, 4, 6syl56 30 . . . 4 x x = y → (z = yx z = y))
8 dvelimhw.1 . . . . 5 (φxφ)
98a1i 10 . . . 4 x x = y → (φxφ))
102, 7, 9hbimd 1815 . . 3 x x = y → ((z = yφ) → x(z = yφ)))
111, 10hbald 1740 . 2 x x = y → (z(z = yφ) → xz(z = yφ)))
12 dvelimhw.2 . . 3 (ψzψ)
13 dvelimhw.3 . . 3 (z = y → (φψ))
1412, 13equsalhw 1838 . 2 (z(z = yφ) ↔ ψ)
1514albii 1566 . 2 (xz(z = yφ) ↔ xψ)
1611, 14, 153imtr3g 260 1 x x = y → (ψxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12olem6  1932
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