Theorem hbsb 1333

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 1210	. 2 |- (A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
2		hbsb.1	. . 3 |- (ph -> A.zph)
3	2	hbsb4 1248	. 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 954  [wsbc 1170

This theorem is referenced by:  2sb5rf 1338  2sb6rf 1339  sb10f 1342  2mo 1447  2eu6 1454  hbsbcg 1951  opabsb 2815  isarep1 3577  oprabval4g 4031

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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