Theorem hbs1 2436

Description: The setvar x is not free in [ y / x ] ph when x and y are distinct. (Contributed by NM, 26-May-1993.)

Assertion

Ref	Expression
hbs1	|- ( [ y / x ] ph -> A. x [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem hbs1

Step	Hyp	Ref	Expression
1		axc16 2135	. 2 |- ( A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )
2		hbsb2 2359	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )
3	1, 2	pm2.61i 176	1 |- ( [ y / x ] ph -> A. x [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1481   [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-12 2047  ax-13 2246

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

This theorem is referenced by:  nfs1v  2437  hbab1  2611  sb5ALT  38731  2sb5ndVD  39146  sb5ALTVD  39149  2sb5ndALT  39168