Metamath Proof Explorer


Theorem nfsb4

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. Usage of this theorem is discouraged because it depends on ax-13 . Theorem nfsb replaces the distinctor with a disjoint variable condition. Visit also nfsbv for a weaker version of nfsb not requiring ax-13 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis nfsb4.1 z φ
Assertion nfsb4 ¬ z z = y z y x φ

Proof

Step Hyp Ref Expression
1 nfsb4.1 z φ
2 nfsb4t x z φ ¬ z z = y z y x φ
3 2 1 mpg ¬ z z = y z y x φ