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Theorem ax12inda 34904
Description: Induction step for constructing a substitution instance of ax-c15 34845 without using ax-c15 34845. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 34903 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 2072 . . 3 𝑤 𝑤 = 𝑦
2 ax12inda.1 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))
32ax12inda2 34903 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))))
4 dveeq2-o 34889 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
54imp 395 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ∀𝑥 𝑤 = 𝑦)
6 hba1-o 34853 . . . . . . . . . 10 (∀𝑥 𝑤 = 𝑦 → ∀𝑥𝑥 𝑤 = 𝑦)
7 equequ2 2123 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
87sps-o 34864 . . . . . . . . . 10 (∀𝑥 𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
96, 8albidh 1963 . . . . . . . . 9 (∀𝑥 𝑤 = 𝑦 → (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑥 𝑥 = 𝑦))
109notbid 309 . . . . . . . 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 473 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (𝑥 = 𝑤𝑥 = 𝑦))
138imbi1d 332 . . . . . . . . . . 11 (∀𝑥 𝑤 = 𝑦 → ((𝑥 = 𝑤 → ∀𝑧𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧𝜑)))
146, 13albidh 1963 . . . . . . . . . 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 331 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑)) ↔ (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
1712, 16imbi12d 335 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((𝑥 = 𝑤 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))) ↔ (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
1811, 17imbi12d 335 . . . . . 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 224 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
2019ex 401 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))))
2120exlimdv 2028 . . 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 197  wa 384  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-c5 34839  ax-c4 34840  ax-c7 34841  ax-c10 34842  ax-c11 34843  ax-c9 34846  ax-c16 34848
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879
This theorem is referenced by: (None)
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