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Theorem ax12indn 33040
Description: Induction step for constructing a substitution instance of ax-c15 32986 without using ax-c15 32986. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12indn.1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Assertion
Ref Expression
ax12indn (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))

Proof of Theorem ax12indn
StepHypRef Expression
1 19.8a 2039 . . 3 ((𝑥 = 𝑦 ∧ ¬ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑))
2 exanali 1773 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
3 hbn1 2007 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
4 hbn1 2007 . . . . 5 (¬ ∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥 ¬ ∀𝑥(𝑥 = 𝑦𝜑))
5 ax12indn.1 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
6 con3 148 . . . . . . 7 ((𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) → (¬ ∀𝑥(𝑥 = 𝑦𝜑) → ¬ 𝜑))
75, 6syl6 34 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦𝜑) → ¬ 𝜑)))
87com23 84 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 → ¬ 𝜑)))
93, 4, 8alrimdh 1777 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
102, 9syl5bi 231 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
111, 10syl5 33 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 ∧ ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
1211expd 451 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by:  ax12indi  33041
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