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Metamath Proof Explorer

Theorem nfcvf

Description: If x and y are distinct, then x is not free in y . Usage of this theorem is discouraged because it depends on ax-13 . See nfcv for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-ext . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)

Ref Expression
Assertion nfcvf ⊢ ¬ ∀ x x = y → Ⅎ _ x y

Proof

Step Hyp Ref Expression
1 nfv ⊢ Ⅎ w ¬ ∀ x x = y
2 nfv ⊢ Ⅎ x w ∈ z
3 elequ2 ⊢ z = y → w ∈ z ↔ w ∈ y
4 2 3 dvelimnf ⊢ ¬ ∀ x x = y → Ⅎ x w ∈ y
5 1 4 nfcd ⊢ ¬ ∀ x x = y → Ⅎ _ x y