Theorem hbsb 1315

Description: If z is not free in ph, it is not free in [y / x]ph when y and z are distinct.

Hypothesis

Ref	Expression
hbsb.1	|- (ph -> A.zph)

Assertion

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

Distinct variable group:   y,z

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		ax-16 1194	. 2 |- (A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
2		hbsb.1	. . 3 |- (ph -> A.zph)
3	2	hbsb4 1232	. 2 |- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
4	1, 3	pm2.61i 126	1 |- ([y / x]ph -> A.z[y / x]ph)

Syntax hints:   -> wi 3  A.wal 950  [wsbc 1153

This theorem is referenced by:  2sb5rf 1320  2sb6rf 1321  sb10f 1324  2mo 1424  2eu6 1431  hbsbcg 1922  opabsb 2777  isarep1 3517  oprabval4g 3970

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155

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