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Theorem ax11inda 2200
 Description: Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. (When z and y are distinct, ax11inda2 2199 may be used instead to avoid the dummy variable w in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11inda.1 x x = w → (x = w → (φx(x = wφ))))
Assertion
Ref Expression
ax11inda x x = y → (x = y → (zφx(x = yzφ))))
Distinct variable groups:   φ,w   x,w   y,w   z,w
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem ax11inda
StepHypRef Expression
1 a9ev 1656 . . 3 w w = y
2 ax11inda.1 . . . . . . 7 x x = w → (x = w → (φx(x = wφ))))
32ax11inda2 2199 . . . . . 6 x x = w → (x = w → (zφx(x = wzφ))))
4 dveeq2-o 2184 . . . . . . . . 9 x x = y → (w = yx w = y))
54imp 418 . . . . . . . 8 ((¬ x x = y w = y) → x w = y)
6 hba1-o 2149 . . . . . . . . . 10 (x w = yxx w = y)
7 equequ2 1686 . . . . . . . . . . 11 (w = y → (x = wx = y))
87sps-o 2159 . . . . . . . . . 10 (x w = y → (x = wx = y))
96, 8albidh 1590 . . . . . . . . 9 (x w = y → (x x = wx x = y))
109notbid 285 . . . . . . . 8 (x w = y → (¬ x x = w ↔ ¬ x x = y))
115, 10syl 15 . . . . . . 7 ((¬ x x = y w = y) → (¬ x x = w ↔ ¬ x x = y))
127adantl 452 . . . . . . . 8 ((¬ x x = y w = y) → (x = wx = y))
138imbi1d 308 . . . . . . . . . . 11 (x w = y → ((x = wzφ) ↔ (x = yzφ)))
146, 13albidh 1590 . . . . . . . . . 10 (x w = y → (x(x = wzφ) ↔ x(x = yzφ)))
155, 14syl 15 . . . . . . . . 9 ((¬ x x = y w = y) → (x(x = wzφ) ↔ x(x = yzφ)))
1615imbi2d 307 . . . . . . . 8 ((¬ x x = y w = y) → ((zφx(x = wzφ)) ↔ (zφx(x = yzφ))))
1712, 16imbi12d 311 . . . . . . 7 ((¬ x x = y w = y) → ((x = w → (zφx(x = wzφ))) ↔ (x = y → (zφx(x = yzφ)))))
1811, 17imbi12d 311 . . . . . 6 ((¬ x x = y w = y) → ((¬ x x = w → (x = w → (zφx(x = wzφ)))) ↔ (¬ x x = y → (x = y → (zφx(x = yzφ))))))
193, 18mpbii 202 . . . . 5 ((¬ x x = y w = y) → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
2019ex 423 . . . 4 x x = y → (w = y → (¬ x x = y → (x = y → (zφx(x = yzφ))))))
2120exlimdv 1636 . . 3 x x = y → (w w = y → (¬ x x = y → (x = y → (zφx(x = yzφ))))))
221, 21mpi 16 . 2 x x = y → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
2322pm2.43i 43 1 x x = y → (x = y → (zφx(x = yzφ))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144 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: (None)
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