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Theorem dvelimfALT2 1680
Description: Proof of dvelimf 1873 using dveeq2 1678 (shown as the last hypothesis) instead of ax-12 1383. This shows that ax-12 1383 could be replaced by dveeq2 1678 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
Hypotheses
Ref Expression
dvelimfALT2.1 (φxφ)
dvelimfALT2.2 (ψzψ)
dvelimfALT2.3 (z = y → (φψ))
dvelimfALT2.4 x x = y → (z = yx z = y))
Assertion
Ref Expression
dvelimfALT2 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 dvelimfALT2
StepHypRef Expression
1 ax-17 1400 . . 3 x x = yz ¬ x x = y)
2 hbn1 1524 . . . 4 x x = yx ¬ x x = y)
3 dvelimfALT2.4 . . . 4 x x = y → (z = yx z = y))
4 dvelimfALT2.1 . . . . 5 (φxφ)
54a1i 9 . . . 4 x x = y → (φxφ))
62, 3, 5hbimd 1447 . . 3 x x = y → ((z = yφ) → x(z = yφ)))
71, 6hbald 1361 . 2 x x = y → (z(z = yφ) → xz(z = yφ)))
8 dvelimfALT2.2 . . 3 (ψzψ)
9 dvelimfALT2.3 . . 3 (z = y → (φψ))
108, 9equsalh 1596 . 2 (z(z = yφ) ↔ ψ)
1110albii 1339 . 2 (xz(z = yφ) ↔ xψ)
127, 10, 113imtr3g 193 1 x x = y → (ψxψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1226   = wceq 1228
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-in1 532  ax-in2 533  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234
This theorem is referenced by: (None)
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