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Metamath Proof Explorer

Theorem dvelimdc

Description: Deduction form of dvelimc . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses dvelimdc.1 ⊢ Ⅎ x φ
dvelimdc.2 ⊢ Ⅎ z φ
dvelimdc.3 ⊢ φ → Ⅎ _ x A
dvelimdc.4 ⊢ φ → Ⅎ _ z B
dvelimdc.5 ⊢ φ → z = y → A = B
Assertion dvelimdc ⊢ φ → ¬ ∀ x x = y → Ⅎ _ x B

Proof

Step Hyp Ref Expression
1 dvelimdc.1 ⊢ Ⅎ x φ
2 dvelimdc.2 ⊢ Ⅎ z φ
3 dvelimdc.3 ⊢ φ → Ⅎ _ x A
4 dvelimdc.4 ⊢ φ → Ⅎ _ z B
5 dvelimdc.5 ⊢ φ → z = y → A = B
6 nfv ⊢ Ⅎ w φ ∧ ¬ ∀ x x = y
7 3 nfcrd ⊢ φ → Ⅎ x w ∈ A
8 4 nfcrd ⊢ φ → Ⅎ z w ∈ B
9 eleq2 ⊢ A = B → w ∈ A ↔ w ∈ B
10 5 9 syl6 ⊢ φ → z = y → w ∈ A ↔ w ∈ B
11 1 2 7 8 10 dvelimdf ⊢ φ → ¬ ∀ x x = y → Ⅎ x w ∈ B
12 11 imp ⊢ φ ∧ ¬ ∀ x x = y → Ⅎ x w ∈ B
13 6 12 nfcd ⊢ φ ∧ ¬ ∀ x x = y → Ⅎ _ x B
14 13 ex ⊢ φ → ¬ ∀ x x = y → Ⅎ _ x B