Theorem nfsb4 2390

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
nfsb4.1	|- F/ z ph

Assertion

Ref	Expression
nfsb4	|- ( -. A. z z = y -> F/ z [ y / x ] ph )

Proof of Theorem nfsb4

Step	Hyp	Ref	Expression
1		nfsb4t 2389	. 2 |- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )
2		nfsb4.1	. 2 |- F/ z ph
3	1, 2	mpg 1724	1 |- ( -. A. z z = y -> F/ z [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco2  2415  nfsb  2440