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Theorem nfsb 2184
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 2135 . 2  |-  ( A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
2 nfsb.1 . . 3  |-  F/ z
ph
32nfsb4 2156 . 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 1549   F/wnf 1553   [wsb 1658
This theorem is referenced by:  hbsb  2185  2sb5rf  2193  2sb6rf  2194  sb10f  2198  sb8eu  2298  2mo  2358  2eu6  2365  cbvab  2553  cbvralf  2918  cbvreu  2922  cbvralsv  2935  cbvrexsv  2936  cbvrab  2946  cbvreucsf  3305  cbvrabcsf  3306  cbvopab1  4270  cbvmpt  4291  ralxpf  5011  cbviota  5415  sb8iota  5417  dfoprab4f  6397  cbvriota  6552  mo5f  23964  cbvmptf  24060  2sb5nd  28584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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