Theorem dvelimdf

Description: Deduction form of dvelimf . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 7-Apr-2004) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimdf.1	⊢ Ⅎ 𝑥 𝜑
	dvelimdf.2	⊢ Ⅎ 𝑧 𝜑
	dvelimdf.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
	dvelimdf.4	⊢ ( 𝜑 → Ⅎ 𝑧 𝜒 )
	dvelimdf.5	⊢ ( 𝜑 → ( 𝑧 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	dvelimdf	⊢ ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		dvelimdf.1	⊢ Ⅎ 𝑥 𝜑
2		dvelimdf.2	⊢ Ⅎ 𝑧 𝜑
3		dvelimdf.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4		dvelimdf.4	⊢ ( 𝜑 → Ⅎ 𝑧 𝜒 )
5		dvelimdf.5	⊢ ( 𝜑 → ( 𝑧 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
6	1 3	nfim1	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜓 )
7	2 4	nfim1	⊢ Ⅎ 𝑧 ( 𝜑 → 𝜒 )
8	5	com12	⊢ ( 𝑧 = 𝑦 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
9	8	pm5.74d	⊢ ( 𝑧 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
10	6 7 9	dvelimf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ( 𝜑 → 𝜒 ) )
11		pm5.5	⊢ ( 𝜑 → ( ( 𝜑 → 𝜒 ) ↔ 𝜒 ) )
12	1 11	nfbidf	⊢ ( 𝜑 → ( Ⅎ 𝑥 ( 𝜑 → 𝜒 ) ↔ Ⅎ 𝑥 𝜒 ) )
13	10 12	syl5ib	⊢ ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜒 ) )