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Metamath Proof Explorer

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 imbitrid ⊢ ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜒 ) )