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Theorem nfsb 2142
Description: If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfsb.1  |-  F/ z
ph
Assertion
Ref Expression
nfsb  |-  F/ z [ y  /  x ] ph
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem nfsb
StepHypRef Expression
1 a16nf 2084 . 2  |-  ( A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
2 nfsb.1 . . 3  |-  F/ z
ph
32nfsb4 2114 . 2  |-  ( -. 
A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
41, 3pm2.61i 158 1  |-  F/ z [ y  /  x ] ph
Colors of variables: wff set class
Syntax hints:   A.wal 1546   F/wnf 1550   [wsb 1655
This theorem is referenced by:  hbsb  2143  2sb5rf  2151  2sb6rf  2152  sb10f  2156  sb8eu  2256  2mo  2316  2eu6  2323  cbvab  2505  cbvralf  2869  cbvreu  2873  cbvralsv  2886  cbvrexsv  2887  cbvrab  2897  cbvreucsf  3256  cbvrabcsf  3257  cbvopab1  4219  cbvmpt  4240  ralxpf  4959  cbviota  5363  sb8iota  5365  dfoprab4f  6344  cbvriota  6496  mo5f  23816  cbvmptf  23910  2sb5nd  27990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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