Theorem nfeqf 1958

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

Assertion

Ref	Expression
nfeqf	|- ((-. A.z z = x /\ -. A.z z = y) -> F/z x = y)

Proof of Theorem nfeqf

Step	Hyp	Ref	Expression
1		nfnae 1956	. . 3 |- F/z -. A.z z = x
2		nfnae 1956	. . 3 |- F/z -. A.z z = y
3	1, 2	nfan 1824	. 2 |- F/z(-. A.z z = x /\ -. A.z z = y)
4		ax12o 1934	. . 3 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
5	4	imp 418	. 2 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
6	3, 5	nfd 1766	1 |- ((-. A.z z = x /\ -. A.z z = y) -> F/z x = y)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

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