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Theorem ax12indi 34733
 Description: Induction step for constructing a substitution instance of ax-c15 34678 without using ax-c15 34678. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax12indn.1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
ax12indi.2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))
Assertion
Ref Expression
ax12indi (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))

Proof of Theorem ax12indi
StepHypRef Expression
1 ax12indn.1 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
21ax12indn 34732 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))
32imp 444 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
4 pm2.21 120 . . . . . 6 𝜑 → (𝜑𝜓))
54imim2i 16 . . . . 5 ((𝑥 = 𝑦 → ¬ 𝜑) → (𝑥 = 𝑦 → (𝜑𝜓)))
65alimi 1888 . . . 4 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
73, 6syl6 35 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
8 ax12indi.2 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))
98imp 444 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓)))
10 ax-1 6 . . . . . 6 (𝜓 → (𝜑𝜓))
1110imim2i 16 . . . . 5 ((𝑥 = 𝑦𝜓) → (𝑥 = 𝑦 → (𝜑𝜓)))
1211alimi 1888 . . . 4 (∀𝑥(𝑥 = 𝑦𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
139, 12syl6 35 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜓 → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
147, 13jad 174 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
1514ex 449 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854 This theorem is referenced by: (None)
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