Theorem nfeqf 1958

Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-12o 2142. (Contributed by Mario Carneiro, 6-Oct-2016.)

Assertion

Ref	Expression
nfeqf	⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz x = y)

Proof of Theorem nfeqf

Step	Hyp	Ref	Expression
1		nfnae 1956	. . 3 ⊢ Ⅎz ¬ ∀z z = x
2		nfnae 1956	. . 3 ⊢ Ⅎz ¬ ∀z z = y
3	1, 2	nfan 1824	. 2 ⊢ Ⅎz(¬ ∀z z = x ∧ ¬ ∀z z = y)
4		ax12o 1934	. . 3 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
5	4	imp 418	. 2 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (x = y → ∀z x = y))
6	3, 5	nfd 1766	1 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz x = y)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by:  equvini  1987  equveli  1988  nfsb4t  2080  sbcom  2089  nfeud2  2216