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Metamath Proof Explorer

Theorem ax12indi

Description: Induction step for constructing a substitution instance of ax-c15 without using ax-c15 . Implication case. (Contributed by NM, 21-Jan-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ax12indn.1 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
ax12indi.2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) )
Assertion ax12indi ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ax12indn.1 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
2 ax12indi.2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) )
3 1 ax12indn ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( ¬ 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) ) ) )
4 3 imp ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) ) )
5 pm2.21 ⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
6 5 imim2i ⊢ ( ( 𝑥 = 𝑦 → ¬ 𝜑 ) → ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
7 6 alimi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
8 4 7 syl6 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) )
9 2 imp ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) )
10 ax-1 ⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) )
11 10 imim2i ⊢ ( ( 𝑥 = 𝑦 → 𝜓 ) → ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
12 11 alimi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
13 9 12 syl6 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) )
14 8 13 jad ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) )
15 14 ex ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) ) )