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Theorem ax12inda2 35096
 Description: Induction step for constructing a substitution instance of ax-c15 35038 without using ax-c15 35038. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 35097. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12inda2.1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Assertion
Ref Expression
ax12inda2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem ax12inda2
StepHypRef Expression
1 ax-1 6 . . . . 5 (∀𝑧𝜑 → (𝑥 = 𝑦 → ∀𝑧𝜑))
2 axc16g-o 35083 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ((𝑥 = 𝑦 → ∀𝑧𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
31, 2syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
43a1d 25 . . 3 (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
54a1d 25 . 2 (∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
6 ax12inda2.1 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
76ax12indalem 35094 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
85, 7pm2.61i 177 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-c5 35032  ax-c4 35033  ax-c7 35034  ax-c10 35035  ax-c11 35036  ax-c9 35039  ax-c16 35041 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828 This theorem is referenced by:  ax12inda  35097
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