Theorem hbsb 1332

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 1209	. 2 |- (A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
2		hbsb.1	. . 3 |- (ph -> A.zph)
3	2	hbsb4 1247	. 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 953  [wsbc 1169

This theorem is referenced by:  2sb5rf 1337  2sb6rf 1338  sb10f 1341  2mo 1446  2eu6 1453  hbsbcg 1948  opabsb 2811  isarep1 3573  oprabval4g 4026

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171

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