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Theorem ax11v2 1679
 Description: Recovery of ax11o 1681 from ax11v 1686 without using ax-11 1374. The hypothesis is even weaker than ax11v 1686, with z both distinct from x and not occurring in φ. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1681. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11v2.1 (x = z → (φx(x = zφ)))
Assertion
Ref Expression
ax11v2 x x = y → (x = y → (φx(x = yφ))))
Distinct variable groups:   x,z   y,z   φ,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11v2
StepHypRef Expression
1 a9e 1564 . 2 z z = y
2 ax11v2.1 . . . . 5 (x = z → (φx(x = zφ)))
3 equequ2 1577 . . . . . . 7 (z = y → (x = zx = y))
43adantl 262 . . . . . 6 ((¬ x x = y z = y) → (x = zx = y))
5 dveeq2 1674 . . . . . . . . 9 x x = y → (z = yx z = y))
65imp 115 . . . . . . . 8 ((¬ x x = y z = y) → x z = y)
7 hba1 1411 . . . . . . . . 9 (x z = yxx z = y)
83imbi1d 220 . . . . . . . . . 10 (z = y → ((x = zφ) ↔ (x = yφ)))
98sps 1408 . . . . . . . . 9 (x z = y → ((x = zφ) ↔ (x = yφ)))
107, 9albidh 1345 . . . . . . . 8 (x z = y → (x(x = zφ) ↔ x(x = yφ)))
116, 10syl 14 . . . . . . 7 ((¬ x x = y z = y) → (x(x = zφ) ↔ x(x = yφ)))
1211imbi2d 219 . . . . . 6 ((¬ x x = y z = y) → ((φx(x = zφ)) ↔ (φx(x = yφ))))
134, 12imbi12d 223 . . . . 5 ((¬ x x = y z = y) → ((x = z → (φx(x = zφ))) ↔ (x = y → (φx(x = yφ)))))
142, 13mpbii 136 . . . 4 ((¬ x x = y z = y) → (x = y → (φx(x = yφ))))
1514ex 108 . . 3 x x = y → (z = y → (x = y → (φx(x = yφ)))))
1615exlimdv 1678 . 2 x x = y → (z z = y → (x = y → (φx(x = yφ)))))
171, 16mpi 15 1 x x = y → (x = y → (φx(x = yφ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1224  ∃wex 1358 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-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624 This theorem is referenced by:  ax11a2  1680
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