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Theorem ax12inda 34052
Description: Induction step for constructing a substitution instance of ax-c15 33993 without using ax-c15 33993. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 34051 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
ax12inda.1 (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))
Assertion
Ref Expression
ax12inda (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Distinct variable groups:   𝜑,𝑤   𝑥,𝑤   𝑦,𝑤   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem ax12inda
StepHypRef Expression
1 ax6ev 1888 . . 3 𝑤 𝑤 = 𝑦
2 ax12inda.1 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))
32ax12inda2 34051 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))))
4 dveeq2-o 34037 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
54imp 445 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ∀𝑥 𝑤 = 𝑦)
6 hba1-o 34001 . . . . . . . . . 10 (∀𝑥 𝑤 = 𝑦 → ∀𝑥𝑥 𝑤 = 𝑦)
7 equequ2 1951 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
87sps-o 34012 . . . . . . . . . 10 (∀𝑥 𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
96, 8albidh 1791 . . . . . . . . 9 (∀𝑥 𝑤 = 𝑦 → (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑥 𝑥 = 𝑦))
109notbid 308 . . . . . . . 8 (∀𝑥 𝑤 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑤 ↔ ¬ ∀𝑥 𝑥 = 𝑦))
115, 10syl 17 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (¬ ∀𝑥 𝑥 = 𝑤 ↔ ¬ ∀𝑥 𝑥 = 𝑦))
127adantl 482 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (𝑥 = 𝑤𝑥 = 𝑦))
138imbi1d 331 . . . . . . . . . . 11 (∀𝑥 𝑤 = 𝑦 → ((𝑥 = 𝑤 → ∀𝑧𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧𝜑)))
146, 13albidh 1791 . . . . . . . . . 10 (∀𝑥 𝑤 = 𝑦 → (∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
155, 14syl 17 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
1615imbi2d 330 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑)) ↔ (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
1712, 16imbi12d 334 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((𝑥 = 𝑤 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))) ↔ (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
1811, 17imbi12d 334 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑)))) ↔ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))))
193, 18mpbii 223 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
2019ex 450 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))))
2120exlimdv 1859 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑤 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))))
221, 21mpi 20 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
2322pm2.43i 52 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-c5 33987  ax-c4 33988  ax-c7 33989  ax-c10 33990  ax-c11 33991  ax-c9 33994  ax-c16 33996
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708
This theorem is referenced by: (None)
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