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 ) ) ) ) )